Uncertainty quantification of the pressure waveform using a Windkessel model

Abstract

The Windkessel (WK) model is a simplified mathematical model used to represent the systemic arterial circulation. While the WK model is useful for studying blood flow dynamics, it suffers from inaccuracies or uncertainties that should be considered when using it to make physiological predictions. This paper aims to develop an efficient and easy-to-implement uncertainty quantification method based on a local gradient-based formulation to quantify the uncertainty of the pressure waveform resulting from aleatory uncertainties of the WK parameters and flow waveform. The proposed methodology, tested against Monte Carlo simulations, demonstrates good agreement in estimating blood pressure uncertainties due to uncertain Windkessel parameters, but less agreement considering uncertain blood-flow waveforms. To illustrate our methodology's applicability, we assessed the aortic pressure uncertainty generated by Windkessel parameters-sets from an available in silico database representing healthy adults. The results from the proposed formulation align qualitatively with those in the database and in vivo data. Furthermore, we investigated how changes in the uncertainty of the Windkessel parameters affect the uncertainty of systolic, diastolic, and pulse pressures. We found that peripheral resistance uncertainty produces the most significant change in the systolic and diastolic blood pressure uncertainties. On the other hand, compliance uncertainty considerably modifies the pulse pressure standard deviation. The presented expansion-based method is a tool for efficiently propagating the Windkessel parameters' uncertainty to the pressure waveform. The Windkessel model's clinical use depends on the reliability of the pressure in the presence of input uncertainties, which can be efficiently investigated with the proposed methodology. For instance, in wearable technology that uses sensor data and the Windkessel model to estimate systolic and diastolic blood pressures, it is important to check the confidence level in these calculations to ensure that the pressures accurately reflect the patient's cardiovascular condition.

Keywords: Windkessel model; direct differentiation method; hemodynamics; sensitivity analysis; uncertainty quantification.

FIGURE 1

Verification of the proposed uncertainty quantification method. The first row shows the mean and standard deviation (STD) of the blood flow waveform due to variations of the stroke volume and heart rate STD from 0% to 10% of the mean value, which correspond to those of the 25‐year‐old baseline subject, namely, 66.8 mL and 75 beats/min, respectively. The Windkessel (WK) parameters' standard deviation varies from 0 to 10% of the mean value from the second to the fifth row; the mean values correspond to WK parameters shown in the first row of Table 1. The resulting mean pressure and STD from Monte Carlo simulations are shown in black lines and shadows, respectively. The blue line is the STD obtained with the proposed expansion‐based method (Equation 6). Root means square errors computed with Equation (24) are shown in each plot.

FIGURE 2

Aortic pressure with standard deviations; for (A) 25 and (B) 75 decades. The shaded area shows the standard deviation of pressure obtained with Equation (6). The black line is the pressure for the mean values of the Windkessel parameters. The blue line is the aortic pressure from the in silico database (baseline subject of the decade). The root mean square error with the baseline subject's pressure as a reference is shown in each plot.

FIGURE 3

Systolic blood pressure (SBP), diastolic blood pressure (SBP) and pulse pressure (PP) variations with age, with standard deviations (STD), for (A) the database, (B) in vivo data, and (C) the current method (obtained with the expansion‐based approach). The solid lines indicate mean values, and the dashed lines indicated +/− STD. The in silico and in vivo values were taken from References , , respectively. Column (D) compares the STD for all cases.

FIGURE 4

Standard deviation of (A) systolic (STD SBP) and (B) diastolic (STD DBP) blood pressures as a function of the standard deviation of a Windkessel parameter (${\mathit{WK}}_i=\left(R,C,Z\right)$) normalized by a reference value shown in the first row of Table 1 $\left({\left[\mathit{std}R\right]}_0=0.1622\frac{\text{mmHg}\bullet \mathrm{s}}{\mathrm{mL}},{\left[\mathit{std}C\right]}_0=0.3472\frac{\mathrm{mL}}{\text{mmHg}}\text{and}{\left[\mathit{std}Z\right]}_0=0.0073\text{mmHg}\frac{\text{mmHg}\bullet \mathrm{s}}{\mathrm{mL}}\right).$ Expansion‐based method results (EB) are represented by lines with dots and those obtained with Monte Carlo (MC) simulations with dots.

FIGURE 5

Pulse pressure uncertainty (STD PP) as a function of the standard deviations a Windkessel parameters (${\mathit{WK}}_i=\left(R,C,Z\right)$) normalized by (A) the reference values shown in the first row of Table 1 and (B) 10% of the reference values, that is, ${\left[\mathit{std}R\right]}_0=0.01622\frac{\text{mmHg}\bullet \mathrm{s}}{\mathrm{mL}},{\left[\mathit{std}C\right]}_0=0.03472\frac{\mathrm{mL}}{\text{mmHg}}\text{and}{\left[\mathit{std}C\right]}_0=0.00073\text{mmHg}\frac{\text{mmHg}\bullet \mathrm{s}}{\mathrm{mL}}$. Expansion‐based method results (EB) are represented by lines with dots and those obtained with Monte Carlo (MC) simulations with dots.

FIGURE A1

Aortic pressure with standard deviations; from 35 to 65 decades. The shaded area shows the standard deviation of pressure obtained with Equation (6). The black line is the pressure for the mean values of the Windkessel parameters. The blue line is the aortic pressure from the base subject of the decade. The root mean square error with the baseline subject's pressure as a reference is shown in each plot.