Mathematical simulations of paediatric haemodynamics in isolated ventricular septal defect

Figures

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  Figure 1.. (a) Schematic of the mathematical model. The left atrium (LA), left ventricle (LV), right atrium (RA) and right ventricle (RV) are described by elastance functions and connected to the systemic and pulmonary arteries (SA and PA) and veins (SV and PV) through resistors. The circulation is modelled as a compliance chamber. (b and c) Growth trajectories for female and male individuals (respectively) through 1 year of life based on Centers for Disease Control (CDC) data.

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  Figure 2.. Changes in (a) MAP and (b) mPAP over the first 2 years of life in both females and males. Data from multiple studies [21–29] reproduced in the study by Hiebing et al. [8] are also presented. A subplot of mPAP during the first two weeks of life is also provided in (b). End-diastolic volumes (EDV, (c) and (d)) and end-systolic volumes (ESV, (e) and (f)) are presented over the same time frame as subplots (a) and (b). Data from multiple studies [21–29] reproduced in the study by Hiebing et al. [8] are also presented.

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  Figure 3.. (a–c) Simulations in a typical adult, 70 kg male, including atrial and ventricular pressure–volume loops (a), vascular pressures (b), and valvular flows (c). (d–f) Simulations from the same model but with parametrizations based on allometric and maturation-based scaling laws to a 2-day-old. Note that pulmonary pressures are significantly higher, with smaller cardiac volumes. See figure 1 for abbreviations for the model infrastructure.

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  Figure 4.. Simulations in a 7-day-old paediatric female (a,c) and male (b,d) subject with and without a VSD. Cardiac function (a,b) and valvular flows (c,d) are shown for each sex with no VSD, a small VSD, a medium VSD and a large VSD, corresponding to 0, 1, 3 and 6 mm in diameter, respectively. Subplots (c) and (d) include flow across the VSD, with positive flow indicated by left-to-right shunting.

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  Figure 5.. Metrics of haemodynamic and cardiac function for various VSD sizes (provided as diameter) and PVR magnitudes, presented in Wood units (WU), for a 30-day-old male subject. (a) Ratio of mean pulmonary flow to mean systemic flow (Qp/Qs), with the threshold of 1.5 provided. (b) Mean pulmonary arterial pressure (mPAP), with the threshold of 20 mmHg used to define pulmonary hypertension. (c–f) LV, RV, LA and RA cardiac work calculated from simulated pressure–volume loops.

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  Figure 6.. Ventricular pressure–volume loops as a function of age and VSD severity in a female. Parameters of the model are scaled using allometric and maturation-based approaches as described in the text. (a) Haemodynamics in the LV from 1 day to 2 years of age. VSD diameters used in the simulations are 0, 1, 3 and 6 mm in diameter. (b) Haemodynamics in the RV over the same time span.

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  Figure 7.. Ventricular pressure–volume loops as a function of age and VSD severity with different pulmonary vascular conditions in a female. All parameters of the model are scaled as described in the text except for Rp and Cpa. (a and b) Haemodynamics in the LV and RV from 1 day to 2 years of age under the assumption that Rp and Cpa change until 30 days of age. (c and d) Haemodynamics in the LV and RV from 1 day to 2 years of age under the assumption that Rp and Cpa are fixed at the value obtained at 3 days of age.

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  Figure 8.. Variability in (a) pulmonary vascular resistance (Rp), (b) systemic vascular resistance (Rs), (c) pulmonary vascular compliance (Cpa) and (d) systemic vascular compliance (Csa) when sampling maturation parameters in a female. Note that age on the x-axis is provided on a log-scale for better interpretability. Grey lines indicate 100 realizations, while the solid black lines and dashed red lines indicate the average and two standard deviations from the average, respectively.

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  Figure 9.. Variability in LV (a–c) and RV (d–f) pressure–volume loops for small (a and d), medium (b and e) and large (c and f) VSD sizes, corresponding to 1, 3 and 6 mm diameters in a female. Simulations use the pulmonary and systemic vascular parameter curves from figure 8, as well as uncertain total blood volume. Grey lines indicate 100 realizations, while the solid-coloured lines indicate the average.

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  Figure 10.. Variability in female VSD flow for (a) small, (b) medium and (c) large VSD sizes, corresponding to 1, 3 and 6 mm diameters. Y-axes provide different magnitudes for ease of interpretability. Simulations use the pulmonary and systemic vascular parameter curves provided in figure 8, as well as uncertain total blood volume. Grey lines indicate 100 realizations, while the solid-coloured lines indicate the average.

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  Figure 11.. Variability in female (a) mean systemic arterial pressure (MAP), (b) mean pulmonary arterial pressure (mPAP) and (c) the pulmonary-to-systemic flow ratio (Qp/Qs) as a function of age and VSD size. We use scaled-probability density functions (PDFs), which are scaled such that the mode of each PDF is 1, for the sake of interpretability. The width of the distribution indicates uncertainty in the estimate, and the y-axis provides the magnitude of the quantity. Simulations use the pulmonary and systemic vascular parameter curves and uncertain blood volumes from before.