Mathematical simulations of pediatric hemodynamics in isolated ventricular septal defect

The gist The study combines a lumped parameter model of the cardiovascular system with size-dependent resistance and age-dependent scaling relationships to simulate pediatric hemodynamics in isolated ventricular septal defect (VSD) from one day to two years old, providing insight into changes in hemodynamic conditions with various VSD sizes, vascular parameters, and age.

Mathematical Model

The study utilizes systems of ordinary differential equations (ODEs) that simulate dynamic pressure p(t) (mmHg), volume V(t) (mL), and flow q(t) (mL/s) using reformulations electrical circuit equations. The model includes four heart chambers: the left atrium (LA), left ventricle (LV), right atrium (RA), and right ventricle (RV). It also includes systemic arterial (SA), systemic venous (SV), pulmonary arterial (PA), and pulmonary venous (PV) compartments for the circulation. The two atrioventricular (tricuspid and mitral) and two semilunar (pulmonary and aortic) valves are modeled as diodes, closing when the pressure gradient across the valve is negative. To simulate flow through a VSD, an additional resistor between the LV and RV is included. The flow across the ventricular defect, qVSD, is modeled by reduced order orifice flow qVSD = sign(Δp) ⋅ πrVSD squared √p∗ ⋅ (2Δp/rho). The volume equations for the LV and RV are modified to dVL V/dt = qM,Val − qA,Val − qVSD, and dVR V/dt = qT,Val − qP,Val + qVSD.

Parameterization and Age-Dependent Scaling

The system of ODEs is driven by resistance, compliance, elastance, and timing parameters. The model begins by parameterizing the model for a reference adult of either sex. Adult blood volume is assumed to be 4500 mL in females and 5000 mL in males. Age-dependent parameterization uses allometric and nonlinear scaling of the adult parameters. This scaling is driven by pediatric age-weight data to provide the 25th, 50th, and 75th percentile weight for any age group within the first two years of life. The maturation approach simulates nonlinear changes in pulmonary and systemic arterial parameters during the initial transition of the neonatal and pediatric circulation. The scaled PVR, Rp, and systemic vascular resistance (SVR), Rs, are described as a function of time t in days.

Simulating VSD and Pulmonary Vascular Dysfunction

The study investigates the effects of elevated Rp and reduced Cpa with aging. Two scenarios are investigated regarding pulmonary vascular adaptation. In the first scenario, pulmonary vascular parameters adapt using equation (10) during the first 30 days of life, but Rp and Cpa then remain at the fixed value Rp(30) and Cpa(30), while other parameters adapt with age. In the second case, pulmonary adaptation only occurs up until three days of age, and Rp and Cpa are fixed at Rp(3) and Cpa(3), respectively.

Results on VSD Size and PVR Impact

The simulation results in Figure 5 show six simulated scalar metrics against different PVR magnitudes for a 30-day old Male subject with various VSD sizes. The Qp/Qs magnitude is significantly higher in medium and large VSDs compared to small or no VSD. As PVR magnitude increases, there is a decrease in Qp/Qs such that at 30 WU, even large VSDs fall below the 1.5 ratio used clinically for VSD classification. Larger PVR values increase mPAP, with pulmonary hypertensive conditions defined for mPAP > 20 (provided in Figure 5(b)). Cardiac work is smallest in the case of a large VSD, consistent with the largest left-to-right shunt.

Hemodynamic Changes with Age and VSD Severity

LV pressures naturally rise with increased SVR and reduced systemic vascular compliance, consistent with prior findings. In the presence of a medium- or large-sized VSD, larger LV EDV is observed at later age points. The RV undergoes a rapid transition during the initial stages of life in the absence of a VSD, with significantly larger systolic pressures at one day of age. Medium- and large-sized VSDs cause a delayed decrease in RV pressures, in addition to a rightward shift indicative of higher EDVs.

Uncertainty Analysis

The uncertainty analysis shows that LV volumes are relatively higher at two years of age in large-sized VSDs (Figure 9(c)) with larger pressure-volume loop area. The uncertainty in cardiac dynamics is larger with early age, with relatively large uncertainty at later ages as well for medium- and large-sized VSDs. The Qp/Qs values show more gradual changes in hemodynamics with age.

Conclusion

This study provides a computational platform for examining the effects of small-, medium-, and large-sized VSDs on hemodynamics from one day to two years old. Our results are consistent with prior measurements in neonates and provide an initial study for using computational modeling to understand VSD management. This is a necessary step towards the use of computer models in subject-specific management of isolated VSD and other congenital heart diseases which coincide with VSDs in neonates.

Limitations

One limitation is that this initial study provides evidence that relatively simple hemodynamic models, combined with adaptation laws and VSD flow, can reproduce hemodynamics with similar magnitudes to measured data. We did not explicitly calibrate or validate our model to data obtained across VSD subjects, which is the next necessary step in transitioning our approach to a digital shadow and, eventually, a digital twin. We consider a simplified model of the four heart chambers, though future modeling studies should incorporate biventricular interaction to identify how septal wall motion is augmented with the defect.

Acknowledgements

We would like to thank Ashley A. Hiebing and Colleen Witzenburg for providing data from their original study. The code for this work can be found at https://github.com/mjcolebank/VSD Simulation.

References

The references listed are 1 through 55 on page 28. The paper contains a list of references numbered 1 through 55 on page 28. The reference list is provided at the end of the paper.

How it works

The mathematical model simulates dynamic pressure p(t) (mmHg), volume V(t) (mL), and flow q(t) (mL/s) using reformulations electrical circuit equations.

1. Enhanced pediatric hemodynamic prediction for VSD management by integrating size-dependent resistance and age-dependent scaling relationships into a lumped parameter model. This improved system can predict changes in hemodynamic conditions with various VSD sizes, vascular parameters, and age from one day to two years of life.

2. Simulation of pulmonary vascular dysfunction scenarios to assess the impact of elevated PVR on cardiac metrics. The improved AI system can investigate how impaired PVR adaptation (modeled here as an arrested decline) sustains elevated RV pressures and higher mPAP.

3. Uncertainty quantification for age-dependent parameterization of vascular adaptation rules. This allows the system to analyze how uncertainties in the maturation parameters of equations (10) - (11) and total blood volume may change the trajectory of hemodynamics as a function of age.

4. Population level view of simulated hemodynamics through uncertainty propagation. The system can provide a population level view by sampling adaptation parameters and simulating trajectories, showing how LV pressure uncertainty in medium- and large-sized VSDs (Figure 9(b)-(c)) is relatively smaller than that in small-sized VSDs (Figure 9(a)), while the converse is true about volume variability.

5. Predictive modeling for clinical decision support regarding VSD surgery. The system can assist in the decision to operate by showing how larger defect sizes increase LA pressure and LV stroke volume also and reproducing clinical Qp/Qs ratios consistent with management guidelines.

Computer modeling of the cardiovascular system has potential to revolutionize personalized medical care. This is especially promising for congenital heart defects, such as ventricular septal defect (VSD), a hole between the two ventricles of the heart. However, relatively few studies have built computer models for VSD, nor have they considered how natural adaptation to the cardiovascular system with age might interact with the presence of a small, medium, or large size VSD. Here, we combine a lumped parameter model of the cardiovascular system with two key modeling components: a size-dependent resistance dictating shunt flow between the two ventricles and age-dependent scaling relationships for the systemic and pulmonary circulations. Our results provide insight into changes in hemodynamic conditions with various VSD sizes. We investigate the combined effects of VSD size, vascular parameters, and age, showing distinct differences with these three factors. This study lays the necessary foundation for studying VSD and towards building digital shadows and digital twins for managing VSD in pediatrics.