1 Introduction

Based on the report of National Birth Defect Prevention Network in USA, 2.3 per 10,000 babies born with anatomical malformations such as heart valves atresia, stenosis, under development right ventricle or hypoplastic left ventricle [16]. These children mainly born cyanosed because their oxygenated (red) and deoxygenated (blue) blood are mixed together. Desaturated and sometimes inadequate systemic blood flow is the main consequence of this malformation which without any treatment increases the risk of mortality. Some single-ventricle defects can be treated using three-stage palliative operations from the first few days to the next few years of patient’s life. The first stage of the surgery is called Norwood Procedure, which is performed just a few days after the child’s birth. The objective of the first stage is providing unobstructed blood flow from single ventricle to the systemic circulation. In this stage, the blood flow for pulmonary arteries (PA) is supplied by placing a modified Blalock-Taussig shunt between repaired aorta and pulmonary arteries. After 6 months, when the patient becomes progressively more cyanotic, bidirectional Glenn operation is performed. This operation also reduces the single ventricle’s workload by directly connecting the superior vena cava (SVC) to the pulmonary artery. In other words, using this connection half of the deoxygenated blood is directed to the lungs without passing through the single ventricle. The final operation is performed when the patient’s lungs are matured. In this operation, inferior vena cava (IVC) is disconnected from the heart and instead redirected to the pulmonary arteries. This setup practically directs blood flow from SVC and IVC to directly enter pulmonary arteries without passing through the heart [5].

Most institutions have claimed that 95 % of patients after surgical intervention would probably have their normal life; however, they are susceptible to numerous long-term complications [2].

Numerical and clinical studies have demonstrated that the structure of TCPC has crucial role in energy loss and consequently in health of patients after surgery. Researchers have proved that high energy loss in TCPC has harmful effects on respiratory system. In addition, abnormal shear stress on the vessels contributes in thrombus formation, which cannot be neglected [2]. Some clinical studies such as Krishnakutty et al. [10], Fogel et al. [4] and Hjortdal et al. [8] investigated the difference of hemodynamics of blood flow in different TCPC anatomies.

In recent decades, the developments in computational fluid dynamics (CFD) tools have been provided beneficial means to simulate cardiovascular systems. Numerical researches [3, 11–13] have demonstrated that geometrical characteristics of TCPC are greatly effective in determining the efficacy of the operation. For instance, not only does an anatomy with high pressure drop causes lower oxygen saturation, especially under exercise condition, but also can lead to higher risk of thrombus formation and eventually postoperative mortality. Because of this significant importance of geometry optimization, Pekkan et al. [17] have introduced a new tool which can optimize the geometry more efficiently, compared to existing traditional CAD systems. They have shown that utilization of free-form haptic anatomy editing tool (SURGEM) eliminates the time-consuming phase of making a desired geometry using computer-aided design systems. Thus, patient-specific simulation before the surgical procedure can be extremely valuable in choosing an optimized TCPC geometry.

Investigation on T-shaped geometry demonstrated that the collision of IVC and SVC flows is the most important reason for high energy loss occurring in the TCPC. Marsden et al. [13, 23] proposed that pressure drop and energy loss decreases significantly by utilization of a Y-shaped graft. In this procedure, IVC flow is directed to pulmonary arteries by placing a Y-shaped graft.

Another postoperative complication that these patients are facing is non-pulsatile blood flow in pulmonary arteries, which no practical solution has been found yet. Throckmorton et al. [20] utilized a pump to facilitate the blood flow in circulatory system; however, it is not an interesting solution due to the utilization of an external device in the body. Recently, several studies are advocated to the potential benefits of leaving pulsatile sources of antegrade pulmonary blood flow after bidirectional cavopulmonary anastomosis (BCPA) procedure [1, 7, 24]. Wal et al. [21] represented that by adding accessory pulsatile flow, the mean pulmonary artery pressure and oxygen saturation increase significantly and no pulmonary hypertension develops. They showed that long-term survival following Fontan with accessory pulsatile flow is significantly higher than the conventional Fontan and associates with better pulmonary artery growth. The idea of directing a pulsatile flow through stenosed PA with specific anatomy has been numerically tested by Ghoreyshi et al. [6] and Rajabzadeh et al. [18]. They demonstrated that this new idea increases the pulsatility of the pulmonary blood flow significantly; however, it increases the energy loss in TCPC. In order to reduce the energy loss that imposed by adding the pulsatile flow, Rajabzadeh Oghaz et al. [14] proposed a novel idea in which an elastic membrane is placed at the main pulmonary artery. This membrane fluctuates because of high gradient pressure that is imposed from univentricular heart. They proposed that the membrane can be constructed by sewing the leaflets of repaired pulmonary valve (three cups) together.

In current study, the new idea of adding pulsatile flow is taken to test using both T-shaped and Y-shaped geometry. Because of importance of vessel-wall compliance on the numerical results [15], this study is conducted using FSI (Fluid Structure Interaction) approach. Similar to previous researches [6], flow pulsatility along with energy loss are determined through the simulations.

2 Materials and Methods

2.1 Geometric morphology and material

The geometry of T-shaped model is constructed by MRI images, taken from a 9-year-old girl’s thorax after a successful extra-cardiac Fontan surgery. It should be mentioned that approval of institutional committee of ethics on research is obtained. T-shaped configuration of TCPC is obtained by image processing of sequential MRI images with 6 mm thickness, shown in Fig. 1. In this figure, the process of image segmentation and geometry reconstruction is illustrated. In addition, inlets and outlets are virtually extended to provide the fully developed condition for inlet blood flow. In this study, the extra pulsatile flow is added to the TCPC by virtually connecting the stenosed main pulmonary artery (PA) to the main TCPC geometry. The diameter and angle of the PA are defined, through the usage of anatomical data along with the information presented by [6]. After construction of the MRI-based T-shaped geometry, the computer-generated Y-shaped geometry is constructed with anatomical and physiological considerations, Fig. 1. As mentioned before, the T-shaped geometry is obtained from a real TCPC, while Y-shaped construction is done by constricting main IVC (as Left Arm of Y-graft) and also adding a deviational direction to the RPA (as Right Arm of Y-graft).

Fig. 1
figure 1

Sequence process of construction of T-shaped and Y-shaped geometries, a coronal plane, b axial plane, c sagittal plane, d 3D reconstruction part, e T-shaped geometry, f Y-shaped geometry

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In order to consider the effect of vessels compliance, FSI-simulation is conducted using ADINA-FSI software. The solid geometry of T-shaped and Y-shaped models can be seen in Fig. 2. Mesh independency of the study has been investigated by increasing the number of elements and comparing the outlet velocity flux of LPA. Approximately 300,000 tetrahedral elements for fluid model and 200,000 shell elements for solid model are used, with consideration of computational cost and precision of the simulation. The blood is considered as Newtonian fluid with a density of 1060 kg/m3 and viscosity of 0.0035 Pa.s. The governing equation of FSI boundary condition is explained in the following section.

Fig. 2
figure 2

Structure model, as vessel’s wall, with specific description of different vessels, a T-shaped geometry, b Y-shaped geometry

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Veins and arteries have different mechanical properties. Experimental investigation of Hunter et al. [9] shows that 3.2 mm and 200 kPa are good assumptions for thickness and stiffness of pulmonary arteries, respectively. Similarly, Silver et al. [19] showed that material properties for vena cava can be considered as 1.2 mm for thickness and 978 kPa for stiffness. It is easy to deduce that arteries are more elastic compared to the veins; however, arteries are thicker than veins. Due to these characteristics, our primitive simulations showed that both vena cava and pulmonary arteries approximately have equal deformation. In other word, less stiffness of arteries than vena cava will be amended by its thicker wall. This outcome led us to consider mean mechanical properties for both vena cava and pulmonary arteries, and subsequently, it reduces the complexity of the numerical modeling. The mean thickness and stiffness of solid part are chosen to be 2.2 mm and 590 kPa, respectively.

2.2 Flow simulation and boundary conditions

2.2.1 The inlet and outlet boundary conditions

The inflow boundary condition for IVC and SVC is derived from echocardiography data of the patient. Figure 3 shows the inlet flux for both IVC and SVC during inspiration period of the cardiac cycle. As mentioned before, the fully developed flow condition is set up for both IVC and SVC by extending the entrance length.

Fig. 3
figure 3

The inlet flux for IVC and SVC for acardiac cycle

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The pulsatile flow (PF) from the heart is added to the TCPC flow through stenosed pulmonary artery. In this study, five distinct fluxes varying from 2 to 10 % of systemic flow rate are considered as PA boundary conditions. The inlet flux of PA in all five models is depicted in Fig. 4.

Fig. 4
figure 4

The inlet flux of PA for a cardiac cycle for models 1–5

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In the present work, constant pressure outlet is considered for both LPA and RPA. According to clinical investigations, the flow distribution of IVC and SVC between LPA and RPA has crucial impact on patient’s health and has been always an important issue in Fontan surgeries [2]. Since the current study is focused on oscillation of TCPC flow, the outlet pressures are considered such that the blood flow distributes in pulmonary arteries as reported by Whitehead et al. [22]. They expressed that the average LPA flow portion is about 45 % of total of pulmonary blood flow, and this is due to the smaller size of the left lung in comparison with the right one. Our primitive results showed that this flow distribution would be established by setting RPA/LPA pressures at 2300/2000 kPa in T-shaped geometry and 2500/2200 kPa in Y-shaped geometry.

2.2.2 FSI boundary condition

The FSI boundary is considered for fluid–solid interfaces, where it defines the fluid load on the structure as well as the effects of the structural displacements on the fluid velocity field. Equations (1) and (2) represent the Navier–Stokes and Continuity equations for fluid region, respectively.

$$\rho_{f} \left( {\frac{{\partial \vec{u}_{f} }}{\partial t} + \vec{u}_{f} .\nabla \vec{u}_{f} } \right) + \nabla p = \mu \nabla^{2} \vec{u}_{f}$$
(1)
$$\nabla .\vec{u}_{f} = 0$$
(2)

In these equations, ρ f is the density of the fluid, \(\vec{u}_{f}\) is the fluid velocity, p is the pressure, and μ defines the fluid viscosity. Furthermore, the governing equation of solid region is defined by Eq. (3),

$$\rho_{s} \frac{{\partial^{2} \vec{X}_{s} }}{{\partial t^{2} }} = \nabla .{\varvec{\upsigma}}_{s}$$
(3)

where ρ s is the density of solid,\(\overrightarrow {X}_{s}\) is the displacement of solid elements, and \({\varvec{\upsigma}}_{s}\) is denotation of Cauchy stress.

Fluid and solid domains at the interface are linked to each other by FSI boundary conditions, which are represented by Eqs. (4) and (5), and kinematic and dynamic boundary conditions.

$$\frac{{\partial \vec{X}_{s} }}{\partial t} = \vec{u}_{f} (t)$$
(4)
$$\vec{n}_{f} \,.\,{\varvec{\upsigma}}_{f} = \vec{n}_{s} \,.\,{\varvec{\upsigma}}_{s}$$
(5)

where \(\vec{n}\,_{f}\) and \(\vec{n}\,_{s}\) are the normal vectors of fluid and solid surface at the interface boundary.

2.3 Determination of performance parameter

The main purpose of this study is to investigate the effect of adding pulsatile flow on pulsatility of blood flow in pulmonary arteries. In this purpose, pulsatility index (PI) is defined as a parameter for measuring the intensity of flow pulsation.

$${\text{PI}} = \frac{{Q_{{{\text{MAX}} - }} Q_{\text{MIN}} }}{{Q_{\text{MEAN}} }}$$
(6)

In this equation, Q MAX and Q MIN are maximum and minimum fluxes of passing flow through pulmonary arteries. Furthermore, Q MEAN is the average of blood flow throughout the cardiac cycle.

As mentioned before, TCPC leads both IVC and SVC flows to the pulmonary arteries without passing through the ventricle. Therefore, the blood pressure reduces in pulmonary arteries, which is not suitable for patient’s respiratory system. For this reason, we also computed the energy loss which is another parameter for comparison between different geometries. Total energy in each surface can be calculated by Eq. (7).

$$\dot{E}_{i} = \,\int\limits_{i}^{{}} {\rho_{f} (\frac{p}{{\rho_{f} }} + \frac{1}{2}\,\vec{u}_{f}^{2} )\vec{u}_{f} .\,dA}$$
(7)

This equation expresses the total energy of fluid in each cross section along the blood flow. The first term in parenthesis shows the potential energy per unit mass of fluid \((\frac{p}{{\rho_{f} }})\) and the second term is the kinetic energy per unit mass \((\frac{1}{2}\,\vec{u}_{f}^{2} )\). Since the blood velocity and pressure change across the computational domain, the integration calculates the total energy of fluid in each cross section. Consequently, the rate of total energy loss in the flow field is derived as follows,

$$\begin{aligned} \dot{E}_{\text{loss}} = \int\limits_{\text{IVC}} {\rho_{f} \left( {\frac{p}{{\rho_{f} }} + \frac{1}{2}\vec{u}_{f}^{2} } \right)\vec{u}_{f} .{\text{dA}} + \int\limits_{\text{SVC}} {\rho_{f} \left( {\frac{p}{{\rho_{f} }} + \frac{1}{2}\,\vec{u}_{f}^{2} } \right)\vec{u}_{f} .{\text{dA}} + \int\limits_{\text{PA}} {\rho_{f} \left( {\frac{p}{{\rho_{f} }} + \frac{1}{2}\vec{u}_{f}^{2} } \right)\vec{u}_{f} .{\text{dA}}} } } \hfill \\ - \int\limits_{\text{LPA}} {\rho_{f} \left( {\frac{p}{{\rho_{f} }} + \frac{1}{2}\,\vec{u}_{f}^{2} } \right)\vec{u}_{f} .{\text{dA}} - } \int\limits_{\text{RPA}} {\rho_{f} \left( {\frac{p}{{\rho_{f} }} + \frac{1}{2}\vec{u}_{f}^{2} } \right)\vec{u}_{f} .{\text{dA}}} \hfill \\ \end{aligned}$$
(8)

3 Results

3.1 Pulsatility index (PI) and flow patterns

In this study, blood flow is simulated in two different types of TCPC geometry (MRI-based T-shaped and computer-generated Y-shaped), in which PA is added virtually. Each geometry experiences 5 distinct models in which the average PF increases gradually from 2 to 10 % of systemic flow by the amount of 2 % in each step (Fig. 4). The term ‘without PA’ means the original model where no extra pulsatile flow exists.

Figures 5 and 6 demonstrate the outflow flux of LPA and RPA, respectively. For better comparison between T-shaped and Y-shaped geometry, their results are superimposed. The velocity vectors for both geometries are depicted in Fig. 7. Since pulsatile blood flow increases in different models, three models are depicted in this figure. Models 3 and 5, which contain 6 and 10 % of systemic flow, are compared with the without PA model. The velocity vectors are depicted at a specific time (0.15 s) when the pulsatile blood flow reaches to the maximum intensity.

Fig. 5
figure 5

Blood flow rate in LPA for both T-shaped and Y-shaped

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Fig. 6
figure 6

Blood flow rate in RPA for both T-shaped and Y-shaped

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Fig. 7
figure 7

Velocity vector Plot for T-shaped and Y-shaped geometry at T = 0.15 s. Increasing the intensity of pulsatile blood flow and its impact on deviation of LPA blood flow (For better observation, vectors with magnitudes lower than 0.14 m/s are removed)

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Figure 8 demonstrates the impact of increasing PF on PI value. The results show that by increasing the pulsatile flow, PI value changes in both geometries.

Fig. 8
figure 8

The PI Value of different models for both T-shaped and Y-shaped geometries

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We have plotted wall displacement results in Fig. 9. The maximum displacement occurs in the IVC. This is because of jetting flow from PA which prevents the IVC flow. This results in increasing fluid pressure in IVC and consequently increasing wall displacement. As shown by increasing PF from a to c, the maximum wall displacement has been increased. This could be simply explained by increasing the IVC pressure by increasing the PF.

Fig. 9
figure 9

Displacement magnitude contours at t = 0.15 s; a Y-shaped model without PF, b Y-shaped model 3, c Y-shaped model 5

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3.1.1 Left pulmonary artery

According to Fig. 5, as expected, T-shaped geometry has more LPA flow compared to Y-shaped one in all cases. This phenomenon happens due to the utilization of a Y-shaped graft for leading IVC flow toward RPA in Y-shaped geometry. Similarly, comparison between the main model (without PA) and the other 5 models (with PA) demonstrates that the LPA flow is affected by PF, in both geometries. In other word, collision of PF and LPA flow at the entrance of LPA causes deviation of IVC flow toward the RPA. This phenomenon can be seen in Fig. 7.

Figure 8 states that low inflow of PA in case 1 not only has a small effect on pulsatility of LPA flow, but also is a reason of decreasing the PI value. However, the presence of PA in models 2–5 causes significant changes in PI value as it increases about 100 % in model 5. Comparison between two geometries in Fig. 8 shows that generally adding PF is more effective on T-shaped geometry compare to Y-shaped one. The effectiveness of T-shaped geometry is more pronounced in models 2 and 3 as the rise in PI value for the T-shaped is approximately 15 % more than the Y-shaped geometry.

3.1.2 Right pulmonary artery

According to Fig. 6, it can be seen that pulsatility of RPA flow in both geometries is naturally far more than LPA. This statement is also confirmed by comparing PI values presented in Fig. 8. Comparison of flow rates in RPA between the two types of geometries shows that approximately both have identical PI value. As it can be seen from Fig. 8, the PI value in the RPA increases proportional to the extra flow added by PA. In contrast to LPA where the PI value changes in higher PA inflow rates, the PI value in RPA principally increases by 7 % just by adding 2 % of cardiac output to the TCPC. In addition, Fig. 8 expresses that in cases 2–5 the pulsatility index increases linearly to 64 % in comparison to the without PA model.

3.2 Energy loss in TCPC

Figure 10 is depicted in order to compare energy loss between the two geometries. For better perception, the calculated parameter in this figure is the percentage of energy loss which is imposed by adding PF. We called it percentage of energy loss and defined it by following formula,

$${\text{Percentage}}\;{\text{of}}\;{\text{Energy}}\;{\text{Loss}}\;{\text{in}}\;{\text{model}}\;{\text{i}} = \,\frac{{\dot{E}_{{\bmod {\text{el i}}}} - \dot{E}_{{{\text{without}}\;{\text{PA}}}} }}{{\dot{E}_{{{\text{without}}\;{\text{PA}}}} }} \times 100\;\%$$
(9)

where \(\dot{E}_{{{\text{model}}\,i}}\) is the energy loss in the model i and \(\dot{E}_{{{\text{without}}\;{\text{PA}}}}\) is the energy loss in the without PA model.

Fig. 10
figure 10

Percentage of energy loss by increasing extra pulsatile flow

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Collision of IVC and SVC flows, especially in T-shaped geometry, is the most important factor in energy loss. Entrance of pulsatile flow from cardiac and collision of this flow with LPA blood flow reinforces the energy loss. Likewise, higher pulsatile flow from cardiac increases the energy loss in TCPC. Comparison between geometries shows that utilization of Y-shaped geometry and consequently better combination of IVC and SVC flow reduces the energy loss significantly. For example, energy loss of Model 5 compared to without PA model in T-shaped geometry is 54 %, while it is about 27 % in Y-shaped geometry.

4 Discussion

In this study, the impact of new approach to increase the pulsatility of blood flow in pulmonary arteries of a patient-specific TCPC is investigated. This approach expresses that an extra flow from the single ventricle can be added to the TCPC in order to increase the pulsatile index of the TCPC flow. According to this approach, this extra fluid will be directed to the TCPC through the stenosed main pulmonary artery. In this study, T-shaped and Y-shaped geometries are also compared together. The results show that adding an extra pulsatile blood flow from cardiac increases the PI value significantly in both geometry, but comparison between the two geometries shows no significant difference in PI value. On the other hand, addition of extra pulsatile flow imposes extra energy loss to the system, which makes serious postoperative complications for the patient. The results show that using Y-shaped geometry decreases this energy loss, significantly.

Besides, the comparison between the models without PA, for both T-shaped and Y-shaped geometries, with the results reported by Marsden et al. [13], shows that the general flow pattern of blood flow and average velocity magnitude in fluid domain in our study is in agreement with Marsden results.

4.1 Left pulmonary artery

Starting systolic phase of cardiac cycle and consequently addition of PF causes deviation of IVC toward RPA. This is due to collision of PF with LPA flow at the entrance of LPA. As seen in Fig. 7, this collision generates vortices which act as an obstacle for IVC flow to enter to the LPA. The velocity vectors of TCPC show how much effective is the PF on the skewness of streamlines. The more the PF is, the more the skewness of streamlines is resulted. By addition of PF, a portion of IVC flow deviates from entering to the LPA; however, this portion is partially substituted by the PF from cardiac.

Diastolic phase of cardiac cycle results in cutting off the PF; however, the IVC flow cannot enter to the LPA simultaneously. This is due to the laxation time, which is required for settling down the formed vortices and inertia of the skewed flow. According to Fig. 5, this relaxation time is about 0.1 s.

4.2 Right pulmonary artery

As mentioned before, adding PF to the TCPC causes more deviation of IVC flow to the RPA in both geometries. Furthermore, utilization of Y-graft also increases the RPA flow rate. According to the incompressibility of fluid and mass conservation law, the reduced flow of LPA should be added to the RPA; however, Fig. 6 demonstrates that the velocity profile of RPA does not change significantly. In other words, in comparison with LPA, extra flow added to the RPA is not remarked in RPA velocity profile. This is because of larger area of RPA surface in comparison with LPA. Due to this reason, utilization of Y-shaped graft does not have any significant effect on RPA velocity profile.

4.3 Imposed energy loss

Single-ventricle patients have chance of survival by undergoing sequences of surgeries. These surgeries are finalized by Fontan operation in which combination of poor-oxygenated and rich-oxygenated blood is prevented by direct connection of the IVC and SVC to the pulmonary arteries. Consequently, poor-oxygenated and low pulsatile blood is directed to the pulmonary arteries without being pumped by the heart. Hence, the pressure drop and energy loss in TCPC are a crucial issue that have high influence on the health of respiratory system of the patients. In this study, the impact of directing a pulsatile flow through PA in both T-shaped and Y-shaped geometries is investigated. The results show that, although adding PF significantly increases the PI value, this approach increases energy loss which cannot be ignored. On the other hand, the results show that utilization of Y-shaped geometry reduces the energy loss by preventing of collision of IVC and SVC flows.

4.4 Limitations

However; the focus of this study is on hemodynamic investigation of blood flow, the effect of PF flow on vessel wall cannot be ignored. For example, the jet flow of PF to the LPA wall, as can be seen in Fig. 7, can damage endothelial cells of the LPA and trigger formation of aneurysms on the vessel’s wall. Besides, measurement of blood flow in pulmonary arteries using Doppler echocardiography or PC-MRI can be used as trustworthy validation tool which will increase the reliability of the simulation. Unfortunately, such information was not available in our study.

5 Conclusion

In this study, the hemodynamic of blood flow in a patient-specific TCPC has been investigated using a new proposed approach for increasing the pulsatility of blood flow in pulmonary arteries. In this approach, a pulsatile flow is taken from the cardiac of the patient and added to the TCPC through stenosed PA. This new approach has been tested numerically on two different types of TCPC, T-shaped and Y-shaped. To provide more realistic results, the compliance of vessels has been considered using FSI approach. The results demonstrated that PI value increases significantly by adding pulsatile flow to TCPC in both geometries, but approximately identical in both T-shaped and Y-shaped geometries. Moreover, it has been found that although this approach increases the pulsatility of blood flow in TCPC, it also imposes significant energy loss. The most important outcome of the present study is significant reduction of energy loss using Y-shaped geometry. Consequently, according to this study, Y-shaped geometry can be a beneficial replacement for T-shaped one and would have significant improvement on patient’s health as well. We believe more comprehensive study is necessary to make a solid conclusion about this new procedure; however, this study provides a new inside for solving an old problem.