The one and a half ventricle repair (1.5VR) is widely applied in the treatment of congenital heart disease with abnormal structure and function of the right ventricle like the Pulmonary Atresia with Intact Ventricular Septum (PA with IVS) (Talwar et al., 2018; Wright et al., 2019), Tetralogy of Fallot (ToF) (Talwar et al., 2018) and Ebstein’s Anomaly (EA) (Malhotra et al., 2018; Talwar et al., 2018; Akkaya et al., 2019). The operation consists of the bidirectional cavopulmonary shunt (BCPS) and the correction of the intracardiac malformation, and aims to make the right ventricle bear only blood flow of the lower body; thus, the sum of the left and right ventricle stroke work (Ptotal) won’t change while the right ventricular volume load is reduced (Barron, 2018). This operation maintains a low right atrial pressure, pulsating pulmonary blood flow and adequate blood oxygen during the short/medium term (Bhattarai et al., 2017; Talwar et al., 2018), which reduces the occurrence of poor prognosis such as the biventricular repair induced right heart failure (Talwar et al., 2018) and the single ventricular repair induced circulatory failure (Sharma et al., 2012; Talwar et al., 2018).

However, in patients with pulmonary vascular dysplasia or left ventricular dysfunction, complications such as the superior vena cava hypertension and Cardiac Output (CO) reduction are very likely to happen (Liu et al., 2011). When the superior vena cava pressure (Psvc) is higher than the normal value (4–6 mmHg), an adaptive growth of the azygos vein may occur. In the literature, the internal diameter of the azygos vein was positively correlated with Psvc (Preger et al., 1969; Contou et al., 2014), which would lead to a vascular steal phenomenon in the azygos vein, namely the rate of the blood flow from the superior vena cava to the pulmonary artery will decrease, while the blood flow from the azygos vein back to the right ventricle will increase, as shown in Figure 1. This phenomenon causes an increase in the right ventricular preload after 1.5VR. If the Pulmonary Vascular Resistance (PVR) is further increased, irreversible changes will occur in the pulmonary vascular remodeling (Peng et al., 2016), leading to an increase in the right ventricle afterload. Finally, the circulation function after the 1.5VR surgery will deteriorate.

Fortunately, the occurrence of such adverse events can be avoided with reoperation. The clinical reoperation measures include the return biventricular repair (2VR), the ligation of the azygos vein (1.5VR′), the return single ventricular repair (1.0VR), the artificial tonic solution and many others. The selection of the reoperation in clinical practice generally takes into account the morphological parameters and physiological functions of the left and right ventricles (Shimizu et al., 2010). However, there is still a lack of research on the effect of pulmonary vascular development on the hemodynamics after reoperation. Due to the limited number of patients, different conditions of patients and their right ventricles, the effects of pulmonary vascular development on 2VR, 1.5VR′, and 1.0VR cannot be tested through clinical research. For patients with different pulmonary vascular development, it is difficult to design experiments to reproduce the hemodynamics before and after reoperation.

In order to study various hemodynamic problems associated with congenital heart disease (CHD), a wide range of computational models have been established in the literature [i.e., the lumped parameter models (LPM)]. Zhang et al. (2019) developed a zero-dimensional (0D) open-loop model to analyze the hemodynamics in aneurysms. Smith et al. (2004) constructed the smallest human cardiovascular system (OD model) to successfully capture the physiological changes. Ma et al. (2014) constructed an LPM for the blood circulation system of single ventricular patients to observe the recovery of different stenosis degrees of the pulmonary artery after the Gleen surgery. Zhao et al. (2015) used the 0D/3D (three-dimensional) coupling model to study the hemodynamic effects of the anastomoses in modified Blalock-Taussig shunt. Corsini et al. (2018) focused on coronary perfusion and aortic arch hemodynamics using computational multi-domain modeling.

This work aims to develop a simple and effective LPM closed model, which only includes systemic and pulmonary circulation. According to the 1.5VR surgery characteristics, the systemic circulation is divided into upper and lower body blood circulation. Therefore, this work builds on the modeling experience of the former report. We attempted to use the lumped parameter model (LPM) based on the lumped parameter state-variable equation (Di Molfetta et al., 2015; Duanmu et al., 2018) to quantitatively analyze the hemodynamic differences between three reoperation procedures of a given PVR and their sensitivity to PVR. The results will provide guidance to select the strategy for reoperation procedures after 1.5VR failure.

The equivalent circuit model that was used to simulate the cardiovascular system is shown in Figure 2. We coupled the Time-varying Elastic Cardiac Cavity Model with the three-element Windkessel vascular model (Korakianitis and Shi, 2006a) to construct a mathematical model of the cardiovascular system after reoperation. The mathematical model consists of three main parts: the heart, the systemic circulation and the pulmonary circulation.

It can be seen from Table 2 that the hemodynamic parameters of 2VR at each position simulated under the system parameters listed in Table 1 are consistent with the clinically measured physiological ranges.

Figures 4A–H show the impact of PVR on the stroke work of the left ventricle and right ventricle after each reoperation procedure; the equivalent quantitative results are shown in Figure 5A. Within the range from 1 time to 4 times of PVR, in 2VR, Plv decreased from 1.2 W to 0.81 W, Prv increased from 0.25 W to 0.4 W and Ptotal decreased from 1.45 W to 1.21 W. In 1.5VR′, Plv decreased from 1.03 W to 0.46 W, Prv increased from 0.13 W to 0.15 W and Ptotal decreased from 1.16 W to 0.61 W. In 1.0VR, Plv and Ptotal both decreased from 0.54 W to 0.19 W. Within the range from 1 to 4 times of PVR, the rate of change of Plv and Ptotal of 1.0VR was obviously greater than that of other reoperation producers, while the rate of change of Prv of 2VR was apparently greater than that of 1.5VR′ (Figure 5B). The ventricular EF is closely related to SV, end systolic volume (Vlves) and end diastolic volume (Vlved). Figure 6A shows that with the increase of PVR, LVEF decreased from 58.80% to 57.86% and RVEF decreased from 68.09% to 50.87% in 2VR. In 1.5VR′, LVEF decreased from 57.73% to 53.66%, and RVEF decreased from 61.31% to 43.86%. In 1.0VR, LVEF decreased from 54.93% to 47.28%. As the PVR increased from 1 to 4 times, the changing rate of LVEF in 1.0VR was obviously greater than that of other reoperation producers, while the changing rate of RVEF in 1.5VR′ was apparently greater than that of 2VR. Detailed data can be obtained from Figure 6B.

It can be seen from Figures 7A–D that the CO difference among the three kinds of surgical operations was obvious, and the CO of each kind gradually decreased with the increase of PVR. During the systolic time of the left ventricle (Ts), when PVR gradually increased, mCO of 2VR, 1.5VR′, and 1.0VR decreased by 18.53%, 37.58%, and 48.07%, respectively (Figure 8).

The results for Pa, Psvc and Pivc are shown in Figures 9A–D and Figures 10A–H, respectively. In 1.0VR, their wave was without pulsation when PVR increased. Regardless of the condition of PVR, Psvc was always greater than Pivc, and Pa was equal to Psvc in 1.5VR′ in a cardiac cycle. The influence of PVR on mPsvc/mPivc is shown in Figure 11. The ratio mPsvc\mPivc of 2VR and 1.0VR was 1, but it increased from 3.76 to 6.76 when PVR increased in 1.5VR′.

Figures 12A–H and Figures 13A–D show the impact of PVR on Qsvc, Qivc and Qti, respectively. Qsvc of 1.5VR′ have a negative value in the period of 11.2 s ∼ 11.5 s (Ts), while Qsvc of 2VR have a negative value in the period of 11.9 s ∼ 12.0 s (Td). In 1.0VR, the wave of Qsvc and Qivc was without pulsation when PVR increased. With the increase of PVR, the peak of waveform of Qti changed from “E peak” to “A peak” in 2VR and 1.5VR′. The change in the pressure is the cause of the flow-rate redistribution. Therefore, as PVR increased from 1 to 4 times, mQsvc\mQivc of 1.5VR′ decreased from 0.55 to 0.36, while it was 0.67 in both 2VR and 1.0VR (Figure 11).

1.5VR has been employed as a useful surgery in patients with CHD. The postoperative results of the surgery are highly patient-specific. In this paper, we have established three general LPM models, representing reoperation procedures after the failure of 1.5VR. They are closed-loop models that only consider the systemic and pulmonary circulation, with the exclusion of the local fine renal and liver circulation. The purpose of developing general LPM models is to mimic the postoperative hemodynamics of reoperation on specific PVR. These models are simple and effective. In addition, based on the models of this paper, we can adjust the lumped parameters of each vascular segment such as R, C, L, and others, so that clinical phenomena such as the azygos vein growth and the results of surgical procedures can be reproduced. The study parameters include Plv, Prv, Ptotal, CO, Qti, Pa, Psvc, Pivc, Qsvc, and Qivc.

The ventricular stroke work can directly reflect the function of the ventricle. With the increase of PVR, The value of Prv of 2VR and 1.5VR′ gradually increased. This phenomenon is in accordance with the qualitative results inferred from general cardiovascular physiology. Furthermore, the Plv/Prv of 1.5VR′ exceeded that of 2VR, which is consistent with the purpose of the 1.5VR surgery (Talwar et al., 2018). Considering the changing rates in the ventricular stroke work of the three surgical procedures, Plv and Ptotal of 1.0VR are more sensitive to variations in PVR, while Prv of 2VR is more sensitive to changes in PVR. The ventricular EF also has a higher correlation with the ventricular function. Under different PVR conditions, the LVEF and RVEF of the three surgeries in this paper were all about 50%. This result indicates a better ventricular function under the given PVR (Surkova et al., 2019). As PVR increases, the afterload of the right ventricle increases and EF decreases. The results shown in Figure 6 are consistent with trends that are qualitatively inferred from general cardiovascular physiology. As seen from the factor of the changing rate of LVEF and RVEF, RVEF is more sensitive to the change of PVR in 2VR and 1.5VR′.

Under different PVR conditions, LVEF and RVEF of the three surgeries stay around 50% in this work, while the mCO values of 1.5VR′ and 1.0VR are obviously decreased. In other words, the filling of the left ventricle decreases (Verhoeff and Mitchell, 2017), which is consistent with the clinical practice. In the clinical situation, the patient with chronic thromboembolic pulmonary hypertension will have an increase in PVR, and the function of the right heart is particularly important (Ruigrok et al., 2019); thus, maintaining a normal CO (3.5 L/min–5.5 L/min) is very important for normal blood circulation of the human body.

In turn, changes in the ventricular function affect changes in the pressure and flow in various parts of the body. The waves of Pa, Psvc, Pivc, Qsvc, and Qivc are of no pulsation in 1.0VR, which is determined by the surgical characteristics of 1.0VR (Jarvis et al., 2016) and is consistent with the clinical practice. With the increase of PVR, Pa of all three surgical methods increases, resulting in different degrees of pulmonary hypertension (Palazzini et al., 2017). Psvc in 1.5VR′ is greater than Pivc due to the characteristics of 1.5VR′ (Bhattarai et al., 2017). mPsvc\mPivc of 1.5VR′ is the largest among the three kinds of procedures, that is because in 1.5VR′, the superior vena cava anastomoses to the pulmonary artery, and Psvc increases due to the influence of Pa (Bhattarai et al., 2017). During the period Ts, the right ventricle is in systole, resulting in Ppa greater than Psvc in 1.5VR′, then more blood would flow from the pulmonary artery to the superior vena cava; thus, the waveform of Qsvc is negative during Ts. However, the superior vena cava in 2VR is connected to the right atrium, and the negative waveform of Qivc during the period of Td is due to the active contraction of the right atrium. Moreover, mPsvc\mPivc is the reason for the change of mQvc\mQivc which has the smallest value in 1.5VR′, this will lead to a significant decrease in blood perfusion in the upper body.

In addition, Qti is closely related to the ventricular function. The “E peak” and “A peak” of Qti waveform are generated by the active systole of the right ventricle and right atrium, respectively, so the time of occurrence of the peak is different. With the increase of PVR, the waveform peak of Qti changes from “E peak” to “A peak,” which indicates that PVR can affect the right ventricular function to some extent. In reference (Murayama et al., 2019), the ratio of velocity between “E peak” and “A peak” is used to quantify the ventricular diastolic function.

A major limitation of this study is related to the model we established. The model constructed in this essay is relatively simple and does not take into account the factors of interventricular interaction, valve movement and coronary circulation of the great clinical value, although it effectively captures physiological changes. Some scholars have used computational models to verify the importance of interventricular interaction (Korakianitis and Shi, 2006a; Lin et al., 2019) and valve movement (Korakianitis and Shi, 2006a). However, 1.5VR that is generally selected for patients based on their cardiac appearance is characterized by the right ventricle being much smaller than the left one, so the interventricular interaction can be ignored. The other limitation is the lumped parameter, which is retrieved from the value of normal people in literature. However, the parameter for CHD is different from that of the normal people. Due to the limited cases of patients with reoperation after 1.5VR failure, it is difficult to obtain actual clinical data. It is well-known that the lumped parameters determined by the normal people are somewhat different from those of a patient, but the interaction between the peripheral tissues of the human body is almost the same, so the lumped parameters determined by a normal person can be used for patients. In addition, only PVR changes are allowed in the simulation process, while other parameters remain unchanged. It has been reported that PVR changes are caused by pulmonary vascular histology, and the resulting pulmonary hypertension will also affect the physiological properties of other parts, such as decreased pulmonary vascular compliance, etc. (Dragu et al., 2015). In future studies, we can analyze the preoperative data of more patients to solve this problem.

By constructing the LPM of the reoperation after 1.5VR failure, we investigated the three kinds of surgical procedures, including the 2VR, 1.5VR′, and 1.0VR. Through MATLAB simulation analysis, the hemodynamic differences among the three kinds of surgical procedures are quantitatively revealed at different PVR. In this paper, we mainly focus on some values which are the ventricular stroke work, EF, Cardiac Output and pressure and flow rate of the superior and inferior vena cava, which can directly or indirectly reflect the function of the heart. Using these values, we can describe the hemodynamics of reoperation after 1.5VR failure under different PVR. The ventricular stroke work, EF and mCO of the changing rate are important parameters that can reflect the sensitivity of different surgical procedures to PVR. Comprehensive analyses of the above-mentioned parameters show that PVR has a greater impact on 1.5VR′ and 1.0VR. Therefore, the selection of reoperation strategies should be mainly based on the development of pulmonary vasculature. To be noted, the definite outcomes of related procedures need to be verified by further experiments or clinic data.

All datasets generated for this study are included in the article/supplementary material.

YF designed and simulated the LPM of virtual surgery, performed the data analysis, and prepared the initial draft manuscript under the instruction of AQ. YY and XF provided the support of clinical knowledge. AQ subsequently revised the manuscript. All authors approved the final submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.