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.
Cavopulmonary connections are surgical procedures used to treat a variety of complex congenital heart diseases. Such malformations, in which a patient has only one effective or functional cardiac pumping chamber (univentricular heart), are usually fatal within the first days or months of life unless treated surgically.
These ‘single-ventricle’ patients usually undergo a series of surgeries resulting in a ‘Fontan circulation’  in which the single ventricle pumps blood returning from the lungs to the body, and the blood returning from the body travels passively to the lungs via direct blood vessel connections without a pumping chamber. This configuration, called a total cavopulmonary connection (TCPC) , is the result of a three-staged surgical series performed on single ventricle heart patients. In the TCPC, the superior vena cava (SVC) and the inferior vena cava (IVC) are directly connected to the right pulmonary artery (RPA), with the IVC connected via either an intra-atrial tunnel or extra cardiac conduit. The previous surgical step is the bidirectional cavopulmonary anastomosis (BCPA) that consists of the connection between the SVC and the RPA .
The T-junction configuration of the TCPC leads to unusual haemodynamic conditions that are not known to occur naturally in any region of the circulatory system. Computational modelling, clinical observation and experimental studies have suggested that the geometry of the TCPC plays a key role in energy losses and fluid dynamics [2,4] and previous work has considered both offset and Y-shaped alternative designs [5–7].
For this reason, most in vitro and in silico studies using simplified models have focused on local fluid dynamics, with a primary focus on energy dissipation resulting from different connection geometries. The goal of these studies was the geometrical optimization of the connections, in order to minimize the energy expenditure. Moreover, the caval flow distribution to the pulmonary arteries was investigated as an important parameter for patient outcomes and the development of arteriovenous malformations .
One of the main limitations of current computational fluid dynamics (CFD) models relates to the suitability of the inlet and outlet boundary conditions. Boundary conditions must correctly simulate the upstream and downstream fluid dynamics and control flow distribution among the various branches and vessels . In the last decade, three-dimensional models of the cavopulmonary connection coupled to downstream impedances were adopted to assess the relative role played by local geometry and pulmonary vasculature without directly imposing pulmonary flow splits . Nevertheless, such models result in an open-loop configuration where the inlet boundary conditions (flow or pressure) have to be imposed and the effects on global circulatory parameters cannot be determined.
More recently, this approach was applied to patient-specific models derived from magnetic resonance (MR) images and clinical measurements , with the intention of investigating the fluid dynamics in BCPA [10,11] and TCPC under realistic anatomical and boundary conditions [12–15]. Clinical measurements (e.g. pressures and cardiac output) may be acquired at different times, using multiple methods and under varying and ‘abnormal’ conditions (e.g. supine, resting, sedated and intubated). These occurrences may lead to inconsistencies within the data to be used in simulations. Nevertheless, it is evident that such patient data cannot be available for predictive investigations of future scenarios. In particular, boundary conditions for models simulating post-operative haemodynamics cannot be clinically measured prior to the surgical operation. This results in an important limitation for virtual planning of cardiovascular repairs, since the pre-operative measured flow data are usually adopted as inlet boundary conditions for the post-operative simulations.
The aim and the novelty of this study are to perform a TCPC virtual planning using a closed-loop multi-scale modelling approach. The solution at both inlet and outlet boundaries of the detailed region under investigation is coupled with a lumped parameter model (LPM) of the remaining cardiovascular system. A multi-scale closed-loop approach was recently adopted to simulate haemodynamics in the coronary tree on the basis of a patient-specific three-dimensional model and a typical LPM description [16,17]. In the present work, the detailed region of the connections (BCPA and TCPC) is patient-specific and built from an MR scan; similarly, the LPM is customized using the patient's data pre-operatively collected. Different virtual post-operative scenarios are simulated to demonstrate the potential of this methodology. In particular, three TCPC configurations are investigated under rest and exercise conditions.
2. Material and methods
(a) Clinical data
The patient analysed for this test case was a 4-year-old male patient (Lucile Packard Children's Hospital, Stanford, CA, USA), who had undergone a BCPA. The clinical data used for the present study were acquired a month before he underwent the TCPC. MR digital imaging and communications in medicine images were used to build the BCPA geometry, and the haemodynamic data consisted of MR flow tracings, catheterization-derived pressure tracings and echocardiographic Doppler velocity tracings (figure 1a). All clinical data are listed in table 1.
Patient right pulmonary (RPVR) and left pulmonary (LPVR) and upper body (UBSVR) and lower body (LBSVR) systemic vascular resistances (SVRs) were calculated on the basis of mean values of pressures and flows as follows: 2.1 2.2 2.3 and 2.4 where PSVC, Patrium and Parch represent the SVC pressure, the atrial pressure and the aortic arch pressure, respectively. Qp is the pulmonary (or SVC) flow, Qp−R and Qp−L are the right and left pulmonary flows and Qs is the systemic flow (i.e. cardiac output). For the studied patient, RPVR, LPVR, UBSVR and LBSVR were equal to 8.3, 15, 64.2 and 81.6 mm Hg min l−1 (Woods units, WU; 1 WU=7999.2 Pa s l−1), respectively. Total pulmonary vascular resistance in the studied patient was 5.3 WU and the SVR was 35 WU.
(b) Three-dimensional models of cavopulmonary connections and resistance–compliance–resistance (RCR ) blocks
The BCPA model includes the SVC, the RPA and the left pulmonary artery (LPA) and 22 pulmonary branches, with 10 RPA outlets and 12 LPA outlets. A stenosis in the proximal region of the LPA can also be observed. The three-dimensional model has an SVC inlet diameter of 11.6 mm, while the outlet diameters range between 1.8 and 4.5 mm.
Using virtual surgery techniques with the same software [15,18], three different TCPC configurations were created (figure 1b): a traditional T-junction with no offset between the venae cavae (T-TCPC), a configuration with an offset towards the LPA (O-TCPC) and a Y-graft connection with 15 mm branches (Y-TCPC) [7,19]. For all of the models, the IVC-graft diameter is equal to 20 mm, in agreement with standard surgical practice. It should be noted that the pre-operative geometry was modified to create the T- and O-junction, so that the stenosis in the LPA was alleviated. The stenosis is still present in the Y-graft model, but the IVC bifurcation allows the flow to by-pass it. In all the models, the SVC and pulmonary artery portions are identical in the pre- and post-operative models.
Using Ansys ICEM 12.1 (Ansys Inc., Canonsburg, PA, USA), the models were discretized with tetrahedral elements and a grid sensitivity analysis was carried out, progressively increasing the mesh density. In agreement with preliminary CFD simulations (Fluent 12.1, Ansys Inc.), a mesh size of approximately 640 000 and 720 000 for the BCPA and TCPC models, respectively, was found to achieve mesh independence for the quantities of interest (namely, the pressure drop between the inlets and the outlets), and an acceptable CPU time expenditure.
The patient-specific downstream boundary conditions at the pulmonary artery branch outlets of the three-dimensional models were prescribed using RCR impedance blocks (figure 1c), whose values were estimated to best match available pre-operative clinical data (transpulmonary pressure gradient and right/left lung blood flow split to compute the total resistance for each branch) and morphometric pulmonary artery data (proximal and distal resistances, capacitance)  as described in detail in Troianowski et al. .
(c) Pre-operative lumped parameter model of the patient cardiovascular system
The BCPA network comprises 21 blocks: 13 to represent the systemic circulation, six for the pulmonary circulation and two for each abdominal organ (liver, intestine and kidneys) as depicted in figure 1d. Venous valves were also included in the model by means of diodes in the leg venous blocks. Furthermore, the dependence of the great vein (SVC and IVC) resistances on transmural pressure was taken into account. Here, coronary circulation, gravity and respiratory effects were neglected. The heart was described by two time-varying elastances, a single atrium and a single ventricle, respectively. Their contractions were ruled by two activation functions, an atrial one and a ventricular one, shifted in time. Atrioventricular and aortic valves were described by nonlinear diodes. For more details, refer to Migliavacca et al. .
The parameter assessment of the BCPA model was made by first setting the patient's body surface area (BSA equal to 0.66 m2), heart rate (84 bpm) and RPVR, LPVR, UBSVR and LBSVR values.
The effect of the patient's BSA was included by making use of allometric equations , where model parameters are scaled according to proper powers of the BSA specific for the various vascular districts or organs : 2.5 where x and x0 are the vascular (R, C and L) and heart (Rvalve, Lvalve and elastances) parameters for the patient BSA and for a reference body surface area BSA0, respectively. The exponent b is the ‘scaling factor’ and describes the effect of a change in the body size. x0 values corresponding to an adult subject (BSA0 of 1.8 m2) were derived from Snyder & Rideout . This procedure allowed us to set a model for a generic 0.66 m2 subject, having resistance Ri−BSA and compliance Ci−BSA for the i-block. Table 2 reports the b coefficients for all the adopted lumped parameters.
Moreover, the patient-specific information on vascular resistances (RPVR, LPVR, UBSVR and LBSVR) was used to fine tune the resistive model parameters. Then, the patient-specific compliance values of each vascular district were obtained as follows: 2.6 where Ri indicates the patient-specific resistance evaluated for the i-block. The adopted power (−4/3) accounts for the different dependence of the vascular resistances and compliances on vessels' dimensions (R≈l⋅r−4 and C≈l⋅r3, r and l being the radius and the length of a vessel, respectively ).
At this stage, the pre-operative LPM of the investigated clinical case was completed. Hence, pre-operative pressure and flow tracings were obtained, solving the algebraic-differential equation system describing the cardiovascular network by means of Simnon (version 3.00.011, 1998, SSPA Maritime Consulting, Göteborg, Sweden), with the 4th/5th order Runge–Kutta–Fehlberg numerical method (RKF-4th/5th), and an integration time of 0.0005 s.
(d) Closed-loop multi-scale models
Multi-scale BCPA and TCPC models were built starting from the pre-operative LPM, in which the pulmonary region was substituted by each three-dimensional model and the RCR blocks (pulmonary impedances; figure 1c). The three-dimensional geometry was imported in ANSYS Fluent 12.1 and the three-dimensional–LPM coupling was performed by means of user-defined functions written in C++ language as described in Migliavacca et al. . The exchange of pressure and flow information was made at the three-dimensional domain boundary sections once per time step. The zero-dimensional model passes pressures at the boundaries to the three-dimensional model, where the Navier–Stokes equations are solved. Flow rates are computed at the boundaries of the three-dimensional model, and passed as an input to the zero-dimensional model. The BCPA multi-scale simulations were initialized with values of pressures and flows derived from previous simulations run with the stand-alone LPM. The LPM was in turn initialized with reasonable values of pressures and flows derived from the literature. This leads three-dimensional–zero-dimensional simulations to faster reach periodicity (in three cardiac cycles).
Transient simulations for each model were run on an IntelR Core i7 (3 GHz) processor, with a 64 bit operating system. The ordinary differential equation system resulting from the LPM was solved with the explicit Euler method, using a time step of 0.0005 s. The multi-scale simulations required a CPU time expenditure of approximately 1.5 days per cardiac cycle. The blood was assumed to be a Newtonian fluid, with a viscosity of 0.004 kg m s−1 and a density of 1060 kg m−3. No gravitational effects were taken into account. Rigid walls were assumed for the three-dimensional portion of the model.
The BCPA model was tested in rest conditions, similar to the patient's condition during a clinical examination, whereas rest and exercise conditions were simulated for all the TCPC models, in order to evaluate the performances of the different configurations over a range of conditions. Exercise conditions were obtained by doubling the heart rate, reducing both LPVR and RPVR by 40 per cent , and the whole SVR by 33 per cent . The decrease in SVR was obtained by reducing leg resistances by 85 per cent.
3. Results and discussion
(a) Pre-operative lumped parameter model
After setting the heart rate, the body surface area and the resistances on the basis of the measured PVR and SVR, the LPM of the BCPA circulation produced pressures and flows that match the clinical ones (table 1). Averaged values throughout a cardiac cycle of fluid dynamics variables calculated by model simulations show differences of less than 3 per cent from the corresponding clinical data.
In addition, the temporal flow waveforms are well captured by the model, thus proving the good quality of the model. As an example, the predicted waveforms of the caval (IVC and SVC) and pulmonary (LPA and RPA) flows are reported in figure 2 and compared with corresponding clinical flow waveforms. The different oscillations of flow in the two venae cavae are well predicted and the peculiar shape of the IVC flow (owing to the pressure changes in the atrium) is properly described, even if some discrepancies are visible during atrial contraction (approx. 0.4 s). The differences in the first 0.1 s are probably owing to the heart timing.
(b) Multi-scale bidirectional cavopulmonary anastomosis and total cavopulmonary connection models
Comparing the BCPA multi-scale model and the LPM results, only small differences in mean values of flows are observed (less than 1%, table 1), whereas some variation occurred in the flow oscillations (figure 2). Maximum, minimum and mean values of pressures reasonably reproduce the LPM ones, as illustrated in table 1, with higher discrepancy than flow values, owing to the different conditions under which pressure data were acquired, i.e. mechanical ventilation. These differences are mainly owing to a reduction in the pulmonary vascular compliance since the three-dimensional model is rigid.
Results obtained from all the TCPC multi-scale models in rest conditions show comparable variations in global haemodynamic parameters with respect to the BCPA model. The caval flows decrease approximately 25 per cent, with an evident reduction in IVC pulsatility. Unfortunately, no post-operative data were collected for the investigated patient, owing to the invasiveness of such exams and the absence of necessity for clinical aims. Nevertheless, the predicted changes in caval flows are in a fairly good agreement with a previous study of ours , where MR caval flows were exceptionally acquired both pre- and post-operatively in a patient who underwent a TCPC surgery. Additionally, as the IVC has been disconnected from the contracting right atrium, reduced pulsatility in the IVC waveform is expected post-operatively. Hence, imposing inlet flows in the TCPC models equal to the pre-operatively measured caval flows  would have not been accurate. In a more recent study, Marsden and co-workers  constructed post-operative boundary conditions of an open-loop TCPC configuration using pre-operatively measured IVC flow, but with scaled-down amplitude to reduce cardiac pulsatility based on typical Fontan data. Nevertheless, the mean caval flow rates were preserved from pre-operative to post-operative state. In the present study, the changes in both SVC (decrease of mean flow rate) and IVC (decrease of mean flow rate and pulsatility) were directly evaluated by the closed-loop model, enhancing the importance of adopting such a multi-scale methodology.
In the exercise conditions, as a consequence of the imposed changes in heart rate and vascular resistances, the IVC flows in all the TCPC models are much higher than those at rest condition (approx. 23 versus 9.5 ml s−1), while the SVC flow is only slightly increased (by approx. 20%; table 3). It should be noted that these values are a result of the multi-scale model and they are not assumed as inlet conditions. In this regard, it is also interesting to observe the pre- and post-operative pressure–volume (P–V) loop during a cardiac cycle (figure 3) either at rest or under exercise conditions. The TCPC models report a P–V curve shifted towards lower volumes and pressure suggesting that the final surgery (the TCPC) decreases the heart workload. Moreover, the stroke volume slightly decreases during exercise despite an increase in systolic pressure (CO is higher than at rest since heart rate was doubled).
Comparing the TCPC configurations in terms of local fluid dynamics and energy efficiency, some observations can be outlined. Either at rest or under exercise conditions, mean flow rate values differ by less than 1 per cent. Observing pathlines during peak IVC flow (figure 4), stagnation zones can be seen in the central region of the T-TCPC model, with SVC and IVC flows preferentially directed towards the LPA and RPA, respectively. Minor extension of such regions is shown in the O-TCPC model. For the Y-TCPC model, a wide stagnation zone downstream of the LPA–SVC connection can be detected. Moreover, from a qualitative point of view, IVC pathlines in the Y-TCPC and O-TCPC models are distributed to both lungs. A more balanced distribution of hepatic factor present in the IVC flow could avoid ‘long-term’ generation of arteriovenous malformations.
From table 3, the T-TCPC model reports the lowest energy efficiency if compared with the other models (94.4% at rest and 91.9% at exercise), corresponding to the highest power losses (1.54 mW at rest and 4.80 mW at exercise). The Y-TCPC model showed the highest efficiency under rest (96.4%) and exercise (94.4%) conditions, which is related to lower power loss values (1.09 mW at rest and 3.37 mW at exercise), despite the remaining stenosis.
The performed multi-scale simulations evidenced that, although the local haemodynamics of the three investigated TCPC configurations were quite different, the overall cardio-circulatory behaviour was barely affected. Additionally, the energy efficiency of all three simulated surgeries was close to one. These occurrences, in contrast with the findings of a previous study , suggest that the systemic and pulmonary peripheral resistances probably play a major role in Fontan patients compared with the local resistances created by different TCPC configurations. While previous studies have aimed to improve efficiency to reduce the workload on the heart, this benefit is not apparent based on the results of the present study.
The present study showed that a lumped parameter description of the BCPA circulation, despite not providing local haemodynamic information, can be used as a tool to set up the patient-specific parameters in the analogous multi-scale three-dimensional–zero-dimensional model. It is worth noting that an LPM alone, such as the one used in the present study, does not allow one to represent different three-dimensional TCPC configurations. Therefore, a multi-scale approach is an appropriate choice to predict the surgical outcome of the different geometrical configurations. The results obtained with multi-scale simulations revealed negligible differences among TCPC configurations in global circulatory parameters: the TCPC configuration does not seem to significantly influence the global fluid dynamics of the model. Slight improvements in performance in terms of local fluid dynamics were demonstrated by the Y-TCPC model, with better flow distribution, greater efficiency and minor power losses both at rest and in exercise. Although such results are in line with the recent literature studies [7,19], the enhancement in clinical outcome brought by a Y-graft configuration is not yet demonstrated.
The Y-graft anastomosis requires a more complex surgery; therefore further works are needed to validate results of these simulations with statistical evidence.
Future works on adaptive and autoregulatory mechanisms would make more reasonable the post-operative behaviour of similar patient-specific multi-scale models.
This study was supported by a grant from the Fondation Leducq, Paris, France. A.L.M. and W.Y. were also supported by a Beginning Grant in Aid Award from the American Heart Association, and a Burroughs Wellcome Fund Career Award at the Scientific Interface. This common work was also supported by the INRIA associated team funding. We would like to acknowledge software provided for model construction from the open source Simvascular project at simtk.org. MOCHA Investigators: Edward Bove MD and Adam Dorfman MD (University of Michigan, USA); Andrew Taylor MD, Alessandro Giardini MD, Sachin Khambadkone MD, Marc de Leval MD, Silvia Schievano PhD and T.-Y. Hsia MD (Institute of Child Health, UK); G. Hamilton Baker MD and Anthony Hlavacek (Medical University of South Carolina, USA); Francesco Migliavacca PhD, Giancarlo Pennati PhD and Gabriele Dubini PhD (Politecnico di Milano, Italy); Richard Figliola PhD and John McGregor PhD (Clemson University, USA); Alison Marsden PhD (University of California, San Diego, USA); Irene Vignon-Clementel (National Institute for Research in Computer Science and Control, France).
One contribution of 11 to a Theme Issue ‘Towards the virtual physiological human: mathematical and computational case studies’.
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