Introduction

Most of our knowledge of the fetal circulation as well as the fetal-to-neonatal transition at birth is based on animal studies. The relative inaccessibility of the fetus and the extremely high invasiveness of certain measurements are responsible for the lack of human data. Although the human fetal circulation has been studied extensively by Doppler ultrasound imaging, with emphasis on Doppler flow velocity waveforms and blood flow estimates,1,2,3 most of the information regarding blood pressures and oxygen saturation levels of the fetal circulation, as well as data regarding the immediate events occurring at birth, have been derived from sheep.4,5,6

Multiple mathematical models have been proposed as a tool to study the fetal circulation and the cardiovascular fetal-to-neonatal transition. Sá-Couto et al.7 were the first to shed light on the fetal-to-neonatal transition by means of a lumped parameter model of the cardiovascular system, although this model lacked the process of gas exchange and homeostatic control mechanisms. Yigit et al.8 included a gas exchange in their model to investigate and compare delayed and immediate cord clamping scenarios, yet used an adaptation time of 10 min for all transitional vessels. Similar to the fetal-to-neonatal transition models, the lumped parameter models of the fetal circulation9,10,11,12,13,14,15,16 rarely include oxygen saturation levels and homeostatic control mechanisms. Furthermore, both models either described hemodynamics only in terms of mean values or relied on the time-varying elastance model of myocardial contraction, thereby neglecting the phenomena underlying contraction.

Given the issues mentioned above, the formulation of a physically correct and detailed model of the fetal circulation and its cardiovascular fetal-to-neonatal transition for the first 24 h after birth, able to describe as many clinically relevant phasic hemodynamic waveforms, appears crucial to a better understanding of the functioning of the fetal and early neonatal cardiovascular system and its responses to stress and disease. The aim of this study was, therefore, to adapt the well-validated and versatile CircAdapt multi-scale model of the adult cardiovascular system17,18 to construct a model of the fetal circulation and to examine the normal hemodynamic transitions during the first 24 h after birth. Build on physical and physiological principles, the CircAdapt model of the human four-chamber heart and closed-loop circulation enables fast and realistic simulation of beat-to-beat cardiovascular mechanics and hemodynamics in a wide range of physiological loading conditions and cardiovascular pathologies.17,19,20,21,22

Methods

Multi-scale cardiovascular computational model

We adapted the closed-loop CircAdapt computational model of the adult cardiovascular system17,18,23 (www.circadapt.org) to simulate cardiovascular mechanics and hemodynamics of the term fetal cardiovascular circulation. CircAdapt allows fast beat-to-beat simulation of the human cardiovascular system for both research and educational purposes24 and has been successfully used for simulating various (congenital) cardiovascular diseases like Tetralogy of Fallot,21 Fontan circulation,25 and pediatric pulmonary arterial hypertension.26 Build on physiological and physical principles, the CircAdapt model consists of a network of modules, all representing components of the cardiovascular system, such as atrial and ventricular walls and cavities, cardiac valves, the pericardium, large blood vessels, and systemic and pulmonary circulations. Altogether, they enable real-time simulation of atrial and ventricular pressures and volumes, regional myofiber stress and strain in the cardiac walls, flow across valves, blood pressure wave propagation within the main arteries and veins, and pulmonary and systemic resistances. Mean arterial pressure (MAP) and cardiac output (CO) are regulated by adjusting the systemic peripheral resistance and total circulating blood volume, mimicking homeostatic control mechanisms (Appendix). Detailed theoretical descriptions and validations of the various modules have been published previously.17,18,23 In the next few sections, we will describe the most relevant aspects of the modules used in this study.

The fetal model parameters refer to a full-term fetus, when fetal body weight and length are assumed to be ~3.5 kg and ~49 cm,27 respectively, whereas the neonatal model parameters refer to a newborn infant at 24 h after birth. The level of detail in the present model (e.g., the number of vessels and organs/tissues) was chosen with the objective of describing as many clinically relevant sites of the fetal cardiovascular system as possible. Figure 1a provides a schematic representation of the modelled system.

Fig. 1: The CircAdapt model of the fetal cardiovascular system with predicted combined cardiac output distribution and oxygen saturation levels.
figure 1

a In the general overview, the orange-colored tubes and valves as well as the placental arterio-venous element represent connections that close during the fetal-to-neonatal transition. Simulated cycle-averaged b flow rate distribution as a percentage of combined cardiac output shown in white, together with selected reference values in gray, and c oxygen saturation levels shown in white. The dashed line indicates the midwall volume for the left ventricle (LV) and left atrium (LA) in red, right ventricle (RV) and right atrium (RA) in blue, and septum (S) in green. The LV and RV are mechanically coupled at their junction (◦). The foramen ovale (FO) was modelled as unidirectional valve, whereas the ductus arteriosus (DA) and the ductus venosus (DV) were modelled as inertias without valve leaflets. The arrows indicate the direction of flow, whereas nodes (•) indicate locations of pressure and oxygen saturation calculation. AV aortic valve, BR brachial, CE celiac, CR cerebral, DA ductus arteriosus, DV ductus venosus, FE femoral, FO foramen ovale, LA left atrium, LI liver, LV left ventricle, MV mitral valve, PL placental, PV pulmonic valve, RA right atrium, RE renal, RV right ventricle, S septum, TV, tricuspid valve.

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Cardiac module

The atria and ventricles were modelled as contractile chambers with the one-fiber model, relating local myofiber mechanics to global pump mechanics under the assumption that myofiber tension is homogeneously distributed throughout the cardiac walls28 (Fig. 1a). Mechanical ventricular interaction was established by force equilibrium in the junction of the three ventricular walls.18 The pericardium was modelled as a nonlinearly compliant bag.

In this study, values of ventricular tissue parameters were manually chosen so that CircAdapt reproduced experimental curves on fetal lamb myocardium29,30 and human fetal velocity tracings,31,32,33 as well as ventricular wall masses and dimensions of the full-term human fetus,34,35,36,37 indicating right ventricular dominance in utero.

Valve module

Valve pressure gradients are composed of a Bernoulli resistance and an inertial term.38 Valve orifice areas of circular shape calculated from valve diameter are listed in Table 1. The mitral, tricuspid, aortic, and pulmonary valve were modelled as unidirectional valves, whereas the veno-atrial orifices as well as the ductus arteriosus (DA) and ductus venosus (DV) were modelled as open orifices introducing inertia effects and allowing bidirectional flow.

Table 1 Major input parameter values.
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The foramen ovale (FO) was modelled as a one-way valve, assuming that it only permits right-to-left shunting.39,40

Av impedance module

The terminal end of a tube was coupled to a nonlinear three-element arterial-venous (av) model, consisting of a pressure-dependent resistive wave impedance, arterial compliance, and peripheral resistance.41 An extensive description of the pulmonary and systemic resistances can be found in the Appendix.

The fetal cardiovascular model, as presented in Fig. 1a, consists of seven av elements, representing the vascular bed of tissues and organs. The upper body circulation includes the cerebral (CR; supplying the brains) and the brachial (BR; supplying the muscular tissue of the upper trunk and extremities) circulations. The lower body circulation consists of the celiac (CE; supplying the intestines), renal (RE; supplying the kidneys), hepatic (LI; supplying the liver), iliac (IL; supplying the muscular tissue of the pelvis and lower extremities), and the umbilical-placental (PL; supplying the placenta) circulations. The pulmonary circulation is represented by the av-element PU.

The pulmonary and systemic resistances were set to ensure the same combined CO (CCO) distribution as found in the term human fetus (Table 1).

The geometrical properties (tube lengths) of the large arteries and veins connecting the av elements to other elements or valves were based on Reymond et al.42 and Müller and Toro43 for the adult arterial and venous segments, respectively, and were scaled according to the fetal–adult body proportions and lengths.44

Oxygen distribution

The dynamic behavior of the CircAdapt model allows forward and backward blood flow through all cavities and vessels during the complete cardiac cycle. Mixing of blood oxygen saturation in a blood-filled cavity is determined by the in- and outflow of the cavity and their respective oxygen saturations as follows:

$$\frac{{{\mathrm{d}}{\mathrm{SO}}_{2,{\mathrm{cav}}}}}{{{\mathrm{d}}t}} = \frac{1}{{V_{\mathrm{cav}}\left( t \right)}}\mathop {\sum}\limits_{i = 1}^n {\left( {\max \left( {0,q_i\left( t \right)} \right) \left( {{\mathrm{SO}}_{2,{\mathrm{i}}}\left( t \right) - {\mathrm{SO}}_{2,{\mathrm{cav}}}\left( t \right)} \right)} \right)}$$
(1)

with SO2,cav the oxygen saturation of the cavity, SO2,i the oxygen saturation of the inflow, qi the blood flow rate of the inflow, and n the number of flows entering the cavity with volume Vcav. The preferential streaming pattern of the umbilical venous return through the DV and the FO45 was taken into account (Appendix).

Neglecting the minor effect of dissolved oxygen, the oxygen saturation at the venous end of an av element is dependent on the oxygen consumption as follows:

$${\mathrm{SO}}_{2,{\mathrm{ven}}}\left( t \right) = {\mathrm{SO}}_{2,{\mathrm{art}}}\left( t \right) - \frac{{V{\mathrm{O}}_{2,{\mathrm{av}}} \times 100}}{{q_{\mathrm{av}}(t) \times k{\mathrm{O}}_2 \times {\mathrm{Hb}}}}$$
(2)

with SO2,art the oxygen saturation of the corresponding arterial cavity, VO2,av the oxygen consumption of the av element, qav the flow across the av element, kO2 the capacity of hemoglobin to carry oxygen (set to 1.36 mL O2/g Hb,4,46), and Hb the hemoglobin concentration in blood (set to 16 g/dL.4,46) Oxygen consumption in the systemic and pulmonary tissues was derived from fetal lamb data.4 Oxygen uptake in the placenta was assumed to render an oxygen saturation of 65%.46,47,48,49,50

Transitions at birth

The proposed model was controlled by a time- and event-based script of changes occurring at birth, such as lung aeration and umbilical cord clamping, as well as the changes during the transitory phase up to 24 h after birth (Table 2 and Supplementary Figure S1).

Table 2 Timeline of different parameter changes in the pulmonary, systemic, and placental circulations.
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Lung aeration

With the onset of breathing, blood is being oxygenated in the lungs. The rise in oxygen saturation at the venous end of the pulmonary av element (PUven) (SO2,PUven) was described as:

$$\frac{{{\mathrm{dSO}}_{2,{\mathrm{PUven}}}}}{{{\mathrm{d}}t}} = \, \frac{1}{{V_{\mathrm{PUven}}\left( t \right)}} ( {{r}}({t}) \max \left( {0,q_{{\mathrm{PU}}}\left( t \right)} \right) \left( {0.98 - {\mathrm{SO}}_{2,{\mathrm{PUven}}}(t)} \right) \\ \quad + \min \left( {0,q_{\mathrm{PUvenLA}}\left( t \right)} \right) \left( {{\mathrm{SO}}_{2,{\mathrm{PUven}}}(t) - {\mathrm{SO}}_{2,{\mathrm{LA}}}(t)} \right) )$$
(3)

With r a function describing the fraction of pulmonary blood flow (PBF) that is saturated, VPUven the volume of the venous pulmonary cavity (PUven), qPU the flow through the pulmonary av element (PU), SO2,LA the oxygen saturation of the left atrium (LA), and qPUvenLA the retrograde flow from the LA into the pulmonary vein. Note that the pulmonary oxygen uptake is set to render a pulmonary venous oxygen saturation of 98%.51 The function r(t) in Eq. (4) increases linearly from zero at the onset of breathing to one when the lungs are fully aerated:

$$r\left( t \right) = \left\{ {\begin{array}{*{20}{l}} 0 \hfill & {{\mathrm{if}}\,t \,< \,t_0} \hfill \\ {\frac{{t - t_0}}{{t_{\mathrm{s}}}}} \hfill & {{\mathrm{if}}\,t_0 \le t \le t_{\mathrm{s}}} \hfill \\ 1 \hfill & {{\mathrm{else}}} \hfill \end{array}} \right.$$
(4)

with t0 the onset of breathing and ts the time it takes to achieve fully aerated lungs. The value of ts was set to 1 min, based on pre- and post-ductal oxygen saturation levels measured in the first minute after birth.52,53,54,55,56,57,58,59,60,61

After birth, lung aeration immediately triggers the decrease in pulmonary vascular resistance (PVR) and increase in PBF.62 The initial rapid decrease in PVR is followed by a more gradual decline over the following days and weeks.63,64 An estimated reduction in PVR of 74% was reported after the start of ventilation in lambs.65 Based on lamb PVR4,5,65 and pulmonary arterial pressure data,66,67,68 an exponential function was followed by a linear function to mimic the initial rapid decrease in the first 36 min after birth and the more gradual decline in PVR thereafter:

$${\mathrm{PVR}}\left( t \right) = \left\{ {\begin{array}{*{20}{l}} {{\mathrm{PVR}}_0\left( {1 - 0.42\left( {1 - e^{ - 8t}} \right)} \right)} \hfill & {{\mathrm{if}}\,0 \le t \le 36\,{\mathrm{min}}} \hfill \\ {{\mathrm{PVR}}_0\left( {0.58 - 0.016\left( {t - 0.6} \right)} \right)} \hfill & {{\mathrm{if}}\,36\,{\mathrm{min}} \,<\, t \le 12\,{\mathrm{h}}} \hfill \\ {{\mathrm{PVR}}_0\left( {0.41 - 0.0062\left( {t - 12} \right)} \right)} \hfill & {{\mathrm{if}}\,12\,{\mathrm{h}} \,<\, t \le 24\,{\mathrm{h}}} \hfill \end{array}} \right.$$
(5)

with PVR0 the PVR in the full-term fetus.

Cord clamping

Thirty seconds after birth, umbilical vascular resistance was instantly increased to infinity, mimicking umbilical cord clamping.

DA closure

After birth, the DA gradually constricts, with functional closure occurring between 12 and 48 h after birth in the majority of newborn infants.69,70,71,72,73,74,75,76,77 Based on DA diameter (DAD) data from several studies,72,73,74,78,79 an exponential function was formulated to mimic the gradual constriction:

$${\mathrm{DAD}}\left( t \right) = \frac{1}{2}{\mathrm{DAD}}_0 \times e^{ - 0.43t} + \frac{1}{2}{\mathrm{DAD}}_0 \times e^{ - 0.14t}$$
(6)

with DAD the diameter in meters and t the time in hours. The diameter before birth (DAD0) was set to 0.0050 m.78,80,81 The resistance (DAR) was calculated under the assumption of Poiseuille flow:

$${\mathrm{DAR}}\left( t \right) \cong \frac{1}{{{\mathrm{DAD}}^4(t)}}$$
(7)

Mean arterial pressure

The gradual increase in MAP that occurs after birth was modelled with a linear function:

$${\mathrm{MAP}}\left( t \right) = \left( {\frac{{{\mathrm{MAP}}_t - {\mathrm{MAP}}_0}}{{24}}} \right)t + {\mathrm{MAP}}_0$$
(8)

with t the time in hours, and MAP0 and MAPt the reference and target MAP (Appendix), which were set to 48 and 60 mm Hg, respectively.66,73,82,83,84

Simulation protocol

Cardiac cycle length was set to 0.414 s (heart rate of 145 beats/min). Absolute flow and oxygen consumption of the systemic elements were kept the same throughout the entire fetal-to-neonatal transition, even with loss of umbilical venous return (Supplementary Figure S2). After DA closure at 24 h after birth, left ventricle (LV) and right ventricle (RV) are forced to act in series and, hence, to deliver an output of 250 mL/min/kg each. Major input parameter values are presented in Table 1.

Numerical implementation

The set of differential equations describing pressure, volume, and oxygen saturation were solved with a time step of 0.1 ms. All simulations were implemented in MATLAB 2019a (The MathWorks, Natick, MA) on a standard personal computer with an Intel® CoreTM i7 processor and 16 GB RAM.

Results

Validation

For the validation of the model, simulated CCO and distribution, oxygen saturations, intra-cardiac, and arterial blood pressures, as well as flow velocity waveforms, were compared to target data. Target data were obtained from healthy infants before and after birth, as well as from animal experiments, in case human data were lacking. The next few sections (and Tables 1 and 3) give an overview of the key findings. A more extensive description of the results as well as a comparison of input and output values with target data can be found in Supplemental Material.

Table 3 Major output parameter values for the fetal circulation.
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Fetal endpoint

Figure 1b shows the simulated flow rate distribution as a percentage of CCO, together with selected reference values for direct comparison. The systemic and pulmonary resistances were set to ensure the same CCO distribution as found in the human term fetus.4,78,80,85,86,87 Simulated right-to-left output ratio was 1.4, corresponding well with the reported range 1.0–1.5.30,33,78,80,87,88,89,90,91,92,93

Simulated oxygen saturation levels in the term fetus are shown in Fig. 1c. Due to the scarcity of in utero human fetal blood oxygen saturation measurements, validation of oxygen saturation levels was restricted to umbilical saturation levels obtained in newborn infants directly after birth. Umbilical arterial oxygen saturation of 33% is within 1 SD of human newborn target data46,47,48,49,50 (Table 3). The higher oxygen saturation level in the LV compared to the RV (LV: 37%; RV: 31%, is caused by the abundant volume of oxygenated umbilical blood presented to the FO.4 As a consequence, pre-ductal oxygen saturation levels exceed post-ductal oxygen saturation levels.

Simulated and measured94 flow velocity waveforms of the fetal shunts are presented in Fig. 2. Simulated cardiac and vascular pressure waveforms, as well as flow velocity waveforms across the cardiac valves and volumic changes of the fetal cardiac chambers, are shown in Fig. 3a. Both ventricular SV (LV: 4.3 mL; RV: 6.1 mL) and ejection fraction (LV: 63%; RV: 53%) correspond well with clinical values,35,36,37,78,87,90,95 although RV values are slightly lower than reported values (Table 3). An explanation can be attributed from various timings in gestational age in the reported measurements together with a general decrease in RV dominance and increase in PBF during the last trimester.4 In our study, pulmonary flow fraction (20%) and right-to-left shunting across the FO (21%) correspond with a full-term fetus, and hence may differ from reported measurements, explaining the slightly lower RV values.

Fig. 2: Measured (top row) and simulated (bottom row) fetal shunt flow velocity waveforms.
figure 2

Fetal shunt flow velocity waveforms of the a ductus arteriosus, b foramen ovale, and c ductus venosus for two consecutive heartbeats with cycle time tc. Vertical dashed lines and horizontal black/gray bars indicate opening and closing of the left atrioventricular (AV) and ventriculo-arterial (VA) cardiac valves. AS atrial systole, D peak during diastole, S peak during systole. Doppler waveforms adapted from Shah et al.94.

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Fig. 3: Simulated pressure, flow velocity, and volume waveforms for the fetal and neonatal endpoint.
figure 3

a Fetal endpoint refers to the term human fetus, whereas b neonatal endpoint refers to the 24-h-old neonate when the ductus arteriosus is closed. Simulated waveforms for pressure (top), flow velocity (middle), and volume (bottom) are shown for two consecutive beats with cycle time tc of the left (left panel) and right (right panel) side of the heart. Left ventricular pressure and volume are shown in red, whereas right ventricular pressure and volume are shown in blue. Atrial pressure and volume, as well as atrioventricular flow, are shown in gray. Arterial pressure and ventriculo-arterial flow are shown in black. Vertical dashed lines and horizontal black/gray bars indicate opening and closing of the atrioventricular (AV) and ventriculo-arterial (VA) valves of the corresponding left or right side. E and A denote the early and atrial activation waves of the atrioventricular flow velocity waveform.

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Simulated E and A wave velocities of mitral (E: 0.37 m/s; A: 0.39 m/s) and tricuspid (E: 0.46 m/s; A: 0.51 m/s) valve were in agreement with clinical observations in the healthy term fetus2,31,32,96 (Table 3). The wide variation found in reported values can be explained by differences in e.g. gestational age, CO, heart rate, orifice area, and fetal breathing movements, as well as by the relatively small data sets.

Data regarding fetal cardiac and vascular pressures is limited due to the extremely high invasiveness of the measurements. However, both simulated mean atrial (LA: 4.5 mm Hg; RA: 4.6 mm Hg) and MAPs (aorta: 48 mm Hg (set); pulmonary: 49 mm Hg) are within 1 SD of human97 and animal98,99,100,101 target data.

The high resistance ratio between the pulmonary and systemic circulation is reflected in the direction of flow through the DA, with continuous right-to-left shunting6,102,103 (Fig. 4a). This provides the pulmonary arterial blood flow waveform with unique characteristics in utero; pulmonary arterial blood flow is directed towards the lungs during most of the systole, yet away from the lungs throughout diastole (Fig. 4a).104 This phenomena is consistent with fetal lamb data.6,105

Fig. 4: Simulated blood flow waveforms in the ductus arteriosus (DA) and pulmonary artery (PUart) and pressure waveforms in the aorta (AO) and pulmonary artery (PUart).
figure 4

Waveforms shown for two consecutive heartbeats with cycle time tc in a the fetus, and b 1 min, c 14 min, d 1 h, and e 24 h after birth. Vertical dashed lines and horizontal black/gray bars indicate opening and closing of the left atrioventricular (AV) and ventriculo-arterial (VA) cardiac valves.

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Lung aeration

An overview of the simulated model parameter changes and evolution of the modelled resistances, flow rates, pressures, and oxygen saturation levels from the fetus to the 24-h-old neonate are presented in Fig. 5, together with target data for direct comparison. With the onset of breathing, the PVR starts to decrease (Fig. 5a), resulting in a gradual increase in PBF (Fig. 5c). As lung aeration causes oxygenation of pulmonary venous blood, oxygen saturation levels considerably increase (Fig. 5e).

Fig. 5: Simulated model input (black lines) and output (colored lines) parameter changes and direct comparison with clinical data (markers) for the first 24 h after birth.
figure 5

Included parameters are a resistance, b absolute cardiac output, c mean blood flow as a percentage of the combined cardiac output, d mean blood pressure, and e oxygen saturation. AO aorta, CCO combined cardiac output, DA (net) net ductus arteriosus shunting, DA (R–L) ductus arteriosus right-to-left shunting, DAR ductus arteriosus resistance, DV ductus venosus, FO foramen ovale, LA left atrium, LV left ventricle, MAP mean arterial pressure, PL placental arterio-venous element, PLVR placental vascular resistance, PU pulmonary arterio-venous element, PUart arterial end of the pulmonary arterio-venous element PU, PVR pulmonary vascular resistance, PVReff effective pulmonary vascular resistance, RA right atrium, RV right ventricle, SVR systemic vascular resistance.

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Umbilical cord clamping

With the clamping of the umbilical cord, loss of the low-resistance placental bed primarily leads to ① an immediate increase in systemic vascular resistance (SVR) of ~0.90 mm Hg s/mL (Fig. 5a), ② a decrease in LV CO, RV CO, and CCO of ~ 26%, 28%, and 27%, respectively (Fig. 5b), and ③ a brief decay in pre- and post-ductal oxygen saturation (Fig. 5e).

① The decrease in PVR and increase in SVR lead to a decrease in the pressure gradient between the pulmonary and systemic circulation (Fig. 4b). As a consequence, right-to-left DA flow decreases.

② Atrial and ventricular pressure–volume curves are displayed in Fig. 6. As venous return is not yet restored at 1 min after birth, both RV and LV show a lower workload with an SV of 4.5 mL (fetus: 6.1 mL) and 3.3 mL (fetus: 4.3 mL), respectively.106

Fig. 6: Simulated atrial and ventricular pressure-volume loops at different time points during the transition.
figure 6

Simulated pressure–volume loops of the a left atrium, b right atrium, c left ventricle, and d right ventricle in the fetus, as well as 1 min, 1 h, and 24 h after birth.

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③ With the onset of breathing before umbilical cord clamping and rise in PBF, the sudden loss of oxygen uptake in the placenta is replaced within a minute by the oxygen uptake in the lungs (Fig. 5e). Time to reach 90% was ~6.7 and 9.6 min for the pre- and post-ductal sites, respectively.52,53,54,55,56,57,58,59,60,61 Pre- and post-ductal sites reached normal adult value (98%) at ~12 and 15 min after birth, respectively.52,54,57

Transitory phase (and ductus arteriosus closure)

DA flow becomes bidirectional at ~10 min after birth and mainly left-to-right shortly thereafter (Fig. 5c), due to the continuous decrease in PVR (Fig. 5a). The reversed DA flow is counterproductive, forming a short-circuit loop from the LV to the LA through the pulmonary circulation. FO flow meanwhile decreases (Fig. 5c), due to rising LA pressure, with functional FO closure occurring at 15 min after birth. Hence, LA and LV workload considerably increases (LV SV, 1 h: 8.7 mL; fetus: 4.3 mL), whereas RV workload remains fairly similar to that in the fetus (RV SV, 1 h: 6.0 mL; fetus: 6.1 mL) (Fig. 6).

From 1 h after birth, LV output decreases (Fig. 5b), due to the rise in DA resistance (Fig. 5a) and, hence, decrease in PBF. Furthermore, the difference between aortic and pulmonary arterial pressure increases (Fig. 4e), with mean pulmonary arterial pressure 50% of mean aortic pressure at 24 h after birth (Fig. 5d).66

The pressure, flow velocity, and volume waveforms of the left- and right-sided heart for the neonatal endpoint are displayed in Fig. 3b. The decreased A/E ratio for the mitral valve (fetus: 1.07; 24 h: 0.82) and relatively unchanged A/E ratio for the tricuspid valve (fetus: 1.10; 24 h: 1.12) agree with reported values in healthy neonates at 24 h after birth (mitral valve: 0.79; tricuspid valve: 1.12).107

Decreased mean pulmonary arterial pressure (fetus: 49 mm Hg; 24 h: 30 mm Hg) at near-constant RV output resulted in a leftward shift in its pressure–volume curve (Fig. 6). For the LV, the increase in mean aortic pressure (1 h: 49 mm Hg; 24 h: 60 mm Hg) and decrease in PBF contribute to a substantial decrease in workload (LV SV, 1 h: 8.7 mL; 24 h: 6.0 mL).

Discussion

The CircAdapt computational model of the adult cardiovascular circulation was adapted to simulate the fetal circulation as well as its transition to the cardiovascular system of a 24-h-old neonate. Although many mathematical models have been developed to describe the fetal circulation9,10,11,12,13,14,15,16 or the cardiovascular fetal-to-neonatal transition,7,8 they often lack oxygen saturation levels, homeostatic control mechanisms, phasic hemodynamic signals, or describe the heart with a time-varying elastance model. In this study, we identified relevant changes in flow velocity and pressure waveforms, as well as the distributions of blood flow and oxygen saturation levels throughout the cardiovascular system, while mimicking homeostatic pressure-flow control mechanisms.

Fetal endpoint

High mean flow velocity in the DV

In utero, the majority of the blood effectively bypasses the lungs by right-to-left shunting across the DA and FO, with the majority of the FO blood flow coming from DV blood flow.105 In our simulations, the large pressure drop across the DV and the substantially smaller DV cross-sectional area, compared to the DA and FO cross-sectional areas, contribute to the high mean flow velocity (Fig. 2). We hypothesize that this high kinetic energy enables blood to preferentially pass through the FO.

Inverted mitral and tricuspid flow patterns

In contrast to what is normally found in the adult, the fetal atrioventricular waveform is characterized by a late filling peak (A wave) that is higher than the early filling peak (E wave). Reed et al.2 reported an A/E ratio for the tricuspid valve of 1.20 ± 0.12 at 36–39 weeks gestation (1.10 in our model) and an A/E ratio for the mitral valve of 1.11 ± 0.17 at 36–39 weeks gestation (1.07 in our model) (Fig. 3a).2

Vasoconstriction of fetal vessels

With umbilical cord clamping taking place at 30 s after birth, arterial oxygen saturations showed a brief drop of ~3% caused by loss of the relatively high umbilical venous oxygen saturation and flow rate (Fig. 5). However, the onset of breathing and corresponding increase in arterial oxygen partial pressure may facilitate gradual umbilical arterial constriction, with the cessation of umbilical arterial flow usually within 45 s after birth.108,109 We neglected these effects to achieve a simpler model. The rise in arterial oxygen saturation, furthermore, promotes DA constriction,75,76,77 which is mimicked with an exponential function in our simulation. Hence, implementing vasoconstriction upon increases in arterial oxygen saturation may slightly change blood flows as well as oxygen saturation levels and would additionally allow for the simulation of, for example, delayed cord clamping.

Postnatal time to reach oxygen saturation ≥90% is >5 min

In agreement with observational studies,52,53,54,55,56,57,58,59,60,61,110 our simulation demonstrates that newborns remain poorly saturated immediately after birth, and that, although oxygen saturation levels improve steadily, normal adult values are not to be expected within the first 15 min after birth. Several studies reported time to reach an oxygen saturation ≥90% of >5 min,52,53,54,56,57,58,59 with slightly longer times for post-ductal sites due to right-to-left shunting. The combination of persistent right-to-left atrial and DA shunting, and ventilation–perfusion mismatching most likely explain the inability to saturate arterial blood with oxygen in the first minute after birth.

Increased preload of the left ventricle in the early neonatal period

The simulation showed a decrease in CO with umbilical cord clamping, DA-induced temporal increase in LV output afterwards (LV output, fetus: 623 mL/min; 30 min: 1259 mL/min), as well as a gradual decrease from 1 h onwards. Various studies showed similar trends,71,73,79,82,106,111,112,113 although magnitudes may vary. With heart rate not significantly changed, a combination of increased PBF and a widely patent DA with predominantly left-to-right shunting likely explains the two-fold increase in LV output at 30 min after birth. Hence, the neonatal heart is working at a relatively high level of the Frank–Starling curve in the early neonatal period.

Limitations and future work

Potential limitations of this simulation may result from the lack of sufficient human physiological and experimental data to compare simulation results with target data. This lack of sufficient data is, for example, prominent in our assumption on constant oxygen consumption as well as the homeostatic pressure-flow regulation and FO valve function given their relatively crude way of modelling. PBF may be underestimated in our model due to the lack of LA volume overload-induced left-to-right FO shunting. Despite the use of animal data for the decline in PVR, the simulation results provide a good approximation of the transitory phase in different parts of the simulation. Other limitations may arise from the timing of umbilical cord clamping as well as lack of amnion fluid resorption and biomechanical responses at birth.

Future work will include expansion of the model with vasoconstriction and vasodilatation upon changes in arterial oxygen saturation levels as well as left-to-right atrial shunting due to LA volume overload. Incorporation of congenital heart disease or pathologies that may occur during the transitional period will further enhance this model’s potential educational and scientific research impact. The current version of the CircAdapt model and educational simulation tool can be downloaded free of charge from www.circadapt.org.

Conclusion

The CircAdapt model of the adult cardiovascular circulation was adapted to simulate the hemodynamics of the fetal circulation and the fetal-to-neonatal transition. Simulation results were validated using experimental and clinical data. Fetal and neonatal endpoints, as well as the events occurring during and directly after birth and the transition during the first 24 h after birth, were realistically represented. The model can, therefore, be used as an educational tool to demonstrate the hemodynamics of the complex fetal circulation and fetal-to-neonatal transition. Furthermore, it can serve as a research platform for in silico investigation of fetal-to-neonatal hemodynamic changes under pathological conditions, for example, in the presence of congenital heart diseases.

Fig. 7: (Non-)linear pressure-flow relation for the arterio-venous (av) resistance, assuming the volume term in Eq. (A1) is constant.
figure 7

The nonlinearity of the pressure-flow relation is determined by the exponent kav. A linear relation (kav = 1) was assumed for all systemic elements, due to homeostatic control mechanisms. The relation becomes nonlinear for higher kav values. A nonlinear relation (kav = 2) was assumed for the pulmonary element, due to vessel dilation and/or vessel recruitment in the pulmonary circulation. The setpoint for the peripheral resistance is determined by the reference av pressure drop (pav,0) and flow rate (qav,0). Note that the nonlinear pulmonary peripheral resistance increases with a decrease in current pulmonary flow rate (qav).

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