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. 2024 Dec;23(6):1909-1931.
doi: 10.1007/s10237-024-01875-x. Epub 2024 Jul 29.

Efficient uncertainty quantification in a spatially multiscale model of pulmonary arterial and venous hemodynamics

Affiliations

Efficient uncertainty quantification in a spatially multiscale model of pulmonary arterial and venous hemodynamics

M J Colebank et al. Biomech Model Mechanobiol. 2024 Dec.

Abstract

Pulmonary hypertension (PH) is a debilitating disease that alters the structure and function of both the proximal and distal pulmonary vasculature. This alters pressure-flow relationships in the pulmonary arterial and venous trees, though there is a critical knowledge gap in the relationships between proximal and distal hemodynamics in disease. Multiscale computational models enable simulations in both the proximal and distal vasculature. However, model inputs and measured data are inherently uncertain, requiring a full analysis of the sensitivity and uncertainty of the model. Thus, this study quantifies model sensitivity and output uncertainty in a spatially multiscale, pulse-wave propagation model of pulmonary hemodynamics. The model includes fifteen proximal arteries and twelve proximal veins, connected by a two-sided, structured tree model of the distal vasculature. We use polynomial chaos expansions to expedite sensitivity and uncertainty quantification analyses and provide results for both the proximal and distal vasculature. We quantify uncertainty in blood pressure, blood flow rate, wave intensity, wall shear stress, and cyclic stretch. The latter two are important stimuli for endothelial cell mechanotransduction. We conclude that, while nearly all the parameters in our system have some influence on model predictions, the parameters describing the density of the microvascular beds have the largest effects on all simulated quantities in both the proximal and distal arterial and venous circulations.

Keywords: Hemodynamics; Multiscale modeling; Pulse-wave propagation; Sensitivity analysis; Uncertainty quantification.

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Conflict of interest statement

Declarations Conflict of interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Citation diversity statement In agreement with the editorial from the Biomedical Engineering Society (BMES) (Rowson et al. 2021) on biases in citation practices, we have performed an analysis of the gender and race of our bibliography. This was done manually, though automatic probabilistic tools exist. We recognize existing race and gender biases in citation practices and promote the use of diversity statements like this for encouraging fair gender and racial author inclusion and identifying gaps in scientific representation.Our references contain 16% woman(first)/woman(last), 12% man/woman, 16% woman/man, and 56% man/man. This binary gender categorization is limited in that it cannot account for intersex, nonbinary, or transgender people. In addition, our references contain 0% author of color (first)/author of color(last), 5% white author/author of color, 16% author of color/white author, and 79% white author/white author. Our approach to gender and race categorization is limited in that gender and race are assigned by us based on publicly available information and online media. We look forward to future databases that would allow all authors to self-identify race and gender in appropriately anonymized and searchable fashion and new research that enables and supports equitable practices in science.

Figures

Fig. 1
Fig. 1
Schematic of computational model geometry. a A pulmonary artery inflow profile is provided as a boundary condition to the MPA and drives flow through the fifteen proximal arteries. A left atrial pressure waveform is provided as a pressure boundary condition for four proximal pulmonary veins, which are connected to an additional generation of veins. The proximal arteries and veins are connected by the structured tree model, which includes the distal vasculature. b A pictorial representation of the structured tree model and how the parameters α and β are used to determine vessel radii. Note that there are both arterial and venous structured trees, which have the same geometry. MPA: main pulmonary artery; LSV: left superior vein; LIV: left inferior vein; RSV: right superior vein; RIV: right inferior vein
Fig. 2
Fig. 2
Polynomial chaos expansion accuracy for a set of 100 validation datasets for different training dataset sizes and polynomial order (K). Accuracy in the MPA and four large veins is shown for a pressure, b flow rate, c WSS, and d CS. Note that the y-axis is presented on a log-scale
Fig. 3
Fig. 3
Output uncertainty via the PCEs in the proximal arteries. The average value (black) and one standard deviation from the average (blue) are provided for the a MPA, b LPA, and c RPA. Results show pressure (top row), flow rate (middle row), and WSS (bottom row) uncertainty as a function of time. Realizations from the sampling procedure are shown in dash-dotted lines
Fig. 4
Fig. 4
Output uncertainty via the PCEs in the proximal veins. The average value (black) and one standard deviation from the average (red) are provided for the a LIV, b LSV, c RIV, and d RSV. Results show pressure (top row), flow rate (middle row), and WSS (bottom row) uncertainty as a function of time. Realizations from the sampling procedure are shown in dash-dotted lines
Fig. 5
Fig. 5
Output uncertainty in wave intensities using PCEs. The average values for FCWs (red), FEWs (cyan), BCWs (blue), BEWs (magenta), and one standard deviation from their respective averages (same colors, shaded) are provided for the a first three proximal arteries and b the four large veins. Note that, because wave magnitudes vary substantially with vein location, we provide a zoom in subplot in c for the LSV, and RSV. Realizations from the sampling procedure are shown in dash-dotted lines
Fig. 6
Fig. 6
Generalized Sobol’ indices (Eq. (30)) calculated using the PCE coefficients for pressure, flow rate, WSS, and CS. Both first-order (Si, light gray) and total-order (STi, dark gray) Sobol’ indices are provided in the a proximal arteries and b proximal veins. Each bar height represents the median Sobol’ index for the proximal arteries or veins, while the error bars denote the range of Sobol’ indices found in either proximal vasculature
Fig. 7
Fig. 7
Generalized, second-order Sobol’ indices calculated using the PCE coefficients for pressure, flow rate, WSS, and CS. Values of Sij are provided in the a proximal arteries and b proximal veins. Each bar height represents the median Sobol’ index for the proximal arteries or veins, while the error bars denote the range of Sobol’ indices found in either proximal vasculature
Fig. 8
Fig. 8
Generalized Sobol’ indices (Eq. (30)) calculated using the PCE coefficients for FCWs, FEWs, BCWs, and BEWs. Both first-order (Si, light gray) and total-order (STi, dark gray) Sobol’ indices are provided in the a proximal arteries and b proximal veins. Each bar height represents the median Sobol’ index for the proximal arteries or veins, while the error bars denote the range of Sobol’ indices found in either proximal vasculature
Fig. 9
Fig. 9
Generalized, second-order Sobol’ indices for FCWs, FEWs, BCWs, and BEWs. Values of Sij are provided in the a proximal arteries and b proximal veins. Each bar height represents the median Sobol’ index for the proximal arteries or veins, while the error bars denote the range of Sobol’ indices found in either proximal vasculature
Fig. 10
Fig. 10
Output uncertainty via the PCEs in the distal arteries and veins of one of the structured tree beds. The average value (black) and one standard deviation from the average (blue or red shade) are provided for the a α-pathway and b β-pathway. Results show the pressure, flow rate, WSS, and CS uncertainty over the structured tree. Values on the left-most side of the x-axis correspond to the largest arteries in the structured tree, while values on the right-most side of the x-axis correspond to the largest veins in the structured tree. The dashed black line denotes the transition from arteries to veins in the structured tree. Realizations from the sampling procedure are shown in dash-dotted lines
Fig. 11
Fig. 11
Generalized Sobol’ indices (Eq. (30)) calculated using the PCE coefficients for pressure, flow rate, WSS, and CS across all eight of the structured tree beds. Both first-order (Si, light gray) and total-order (STi, dark gray) Sobol’ indices are provided in the a α arteries, b β arteries, c α veins, and d β veins. Each bar height represents the median Sobol’ index for the distal α and β arteries or veins, while the error bars denote the range of Sobol’ indices found in across the different structured tree beds
Fig. 12
Fig. 12
Realization from the training data that includes the "S1" and "S2" components of the pulmonary venous flow rate. a MPA pressure; b LIV flow rate; c LSV flow rate; d RIV flow rate; e RSV flow rate

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