## Abstract

Pulmonary arterial hypertension (PAH) is characterized by pulmonary vascular remodeling resulting in right ventricular (RV) dysfunction and ultimately RV failure. Mechanical stimuli acting on the vessel walls of the full pulmonary tree have not previously been comprehensively characterized. The goal of this study is to characterize wall shear stress (WSS) and strain in pediatric PAH patients at different stages of disease severity using computational patient-specific modeling. Computed tomography, magnetic resonance imaging and right heart catheterization data were collected and assimilated into pulmonary artery (PA) models for patients with and without PAH. Patients were grouped in three disease severity groups (control, moderate and severe) based on clinical evaluations. A finite element solver was employed to quantify hemodynamics and wall strains. To estimate WSS in the distal small PAs with diameters ranging from 50 to 500 \(\upmu \text {m}\), a morphometric tree model was created, with inputs coming from outlets of the 3D model. WSS in the proximal PAs decreased with disease severity (control 20.5 vs. moderate 15.8 vs. severe 6.3 \(\text {dyn}/\text {cm}^2\), \(p<0.05\)). Oscillatory shear index increased in the main pulmonary artery (MPA) with disease severity (0.13 vs. 0.13 vs. 0.2, \(p>0.05\)). Wall strains measured by the first invariant of Green strain tensor decreased with disease severity (0.16 vs. 0.12 vs. 0.11, \(p>0.05\)). Mean WSS for the distal PAs between 100 and 500 \(\upmu \text {m}\) significantly increased with disease severity (20 vs. 52 vs. 116 \(\text {dyn}/\text {cm}^2\), \(p<0.05\)). In conclusion, 3D flow simulations showed that WSS is significantly decreased in the MPA with disease while the mathematical morphometric model suggested increased WSS in the distal small vessels. Computational models can reveal mechanical stimuli acting on vessel walls that may inform patient risk stratification and flow shear experiments.

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## Acknowledgements

The authors would like to thank Dr. Chiu-Yu Chen, Michelle Ogawa, Sam Craft, Aldine Whitfield and Matthew Irvin for their assistance in data collection. This work was supported in part by the Vera Moulton Wall Center at Stanford University. Melody Dong was supported by an NSF graduate research fellowship. High-performance computing resources were provided by Stanford Research Computing and the NSF XSEDE program Towns et al. (2014).

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## Appendix

### Appendix

### Surrogate model

To match clinical targets, we used optimization to identify appropriate parameters of the LPMs. Since it is expensive to evaluate a 0D–3D simulation, a surrogate model is used for the parameter identification. In the surrogate model (Fig. 10), the 3D PA model is replaced by a resistor and RCR models applied to the 3D outlets are reduced to two RCR model in parallel for the RPA and LPA. By adding two equations for the RCR models, the surrogate model can be described as follows:

where variables *V*, *Q*, *P*, \(\Delta P\) and *L* are volume, flow, pressure, pressure drop across a valve and inductance, respectively, *U* is a constant for atrium pressure, subscripts TC, RA, PV, LA, and PA represent the tricuspid valve, right atrium, right ventricle, pulmonary valve, left atrium and pulmonary artery, respectively, and \(R_p\) and \(R_d\) represent the proximal and distal resistance, respectively, in the RPA (LPA). In the surrogate, the auxiliary functions and valve models were kept unchanged.

A standard Runge–Kutta method can be used to solve the ODEs above to obtain pressures, volumes and flow rates for the RV and PA. A multi-objective optimizer (fgoalattain, MATLAB, The Mathworks Inc., Natick, MA) is used to identify LPM parameters (8 variables for the right heart and 3 variables for the PA) to match catheterization derived RV pressures (systolic and diastolic) and PA pressures (systolic, diastolic and mean) and MRI-derived end diastolic volume (EDV) and end systolic volume (ESV) with equal weights. Typical ranges for optimization parameters and constant LPM parameters are listed in Table 4. Initial values for \(C_1, C_2, C_3\) and \(C_4\) could be estimated by,

where \(P_{\min }\) and \(P_{\max }\) are the maximum and minimum pressures for the RV and \(P_\mathrm{PA}\) is the mean PA pressure. We assign a constant value for the resistor that represents the 3D PA model. The value is chosen such that the RV pressure–volume loops produced by the surrogate and the 0D–3D simulation are close. The initial resistance could be estimated by dividing the pressure drop between the inlet and outlets in the 3D model by the pulmonary flow.

### Volumetric flow comparison

PCMRI was performed in the MPA only in most patients. Although the mean flow rate for the MPA produced by the right heart LPM was tuned to match the MRI measurement, volumetric flow waveforms were not directly applied to the inlet of the 3D model. Figure 11 shows a flow waveform comparison between simulations and MRI measurements available in the MPA, RPA and LPA for two representative patients. Overall, there is good agreement in patients C2 and C5. Major differences are present at the systole. This is because the pulmonary mean flow was matched during the LPM optimization while constraints for matching peak flow were not imposed. In future studies, additional clinical information for peak flow rates and the duration of valve opening can be incorporated into the LPM optimization to achieve better agreement with the MRI data (Table 5).

### Morphometric tree generation

For the sake of completeness, some details of morphometric tree generation for the distal small PAs are provided in this Appendix. Huang et al. (1996) performed a morphometric measurement on a healthy adult lung and a diameter-defined Strahler ordering system was used to represent the pulmonary arterial tree. The order number is increased with increasing branch size and only bifurcation is allowed in a Strahler tree. When two vessels of order *n* and \(\le \)*n* meet, the order confluent vessel is kept to be *n* or \(n+1\) only if its diameter is larger than a defined threshold (Kassab et al. 1993). A tree segment is defined as the vessel between two closest bifurcation points and a tree element is defined as multiple segments with the same order connected in series (Huang et al. 1996). With the ordering system above, Huang et al. (1996) divided the PAs into 15 orders between the central left/right PA and the smallest arterioles. For elements at each order, the mean diameter and length were given based on measurements. To describe the branching pattern, a connectivity matrix is created to provide the number of child elements for a parent element at any order. A simple connectivity matrix was made up for illustrative purposes (Table 6). For an entry *C*(*m*, *n*) at the *m*th row and *n*th column of the connectivity matrix, the value represents the average number of child element of order *m* for an element of order *n*. Therefore, the total number of elements of order *m*, \(N_m\), is given by \(N_m=\sum _{n=m}^k C(m,n)N_{n}\), where *k* is the maximum order number. For a given connectivity matrix, the following rules are used to create a morphometric tree model (Spilker et al. 2007). For a root element of order *n*, first find the number of child elements from order 1 to order *n*. If the number of child elements equals to 1, add the child element to the midpoint of the parent element. If the number of child elements equals to 2, add two child elements to the end of the parent element. If the number of child elements, n, is greater than 2, split the parent element into \(n-1\) segments and add the two largest elements of order \(n-1\) to the outlet of the parent element. The remaining elements are added to the interior bifurcation points such that the larger element is closer to the inlet of the parent element. Then add child elements for newly added elements using the same rule. By repeating this process of child element generation from order *n* to order 1, a morphometric tree is created. At each time, when child elements are created, the number of child elements of order *m* for a parent of order *n* is obtained by rounding down *C*(*m*, *n*) and the remainder is kept to be added to *C*(*m*, *n*) when child elements of order *m* are generated for next parent element of order *n*. Figure 12 illustrates the generation of a morphometric tree based on Table 6. Alternatively, a simpler way to represent the distal vascular tree is to use the structured tree model in which the vessel always bifurcates into two smaller vessels with constant factors (Olufsen 1999; Yang et al. 2016). Compared to the structured tree model, the morphometric tree model is derived from ex vivo measurements and multiple child elements are allowed to connect to the same parent element.

The branching pattern can be modified by scaling the connectivity matrix. To model pulmonary vascular pruning (loss of peripheral vasculature), we uniformly reduce the number of child elements for vessels smaller than order 8, i.e., baseline *C*(*m*, *n*) was scaled by \(\alpha \) if \(n \le 8\). Similarly, diameters for order 1–8 were uniformly scaled by \(\beta \) to model wall thickening. By introducing two parameters, \(\alpha \) and \(\beta \), we can modify the structure of a morphometric tree. An optimization is performed to identify a set of \(\alpha \) and \(\beta \) such that the morphometric tree resistance matches a target resistance value that is calculated by dividing the pressure drop between the 3D outlets and capillaries by mean outlet flow.

When a morphometric tree is created, the resistance is calculated using an electric analogy. A bifurcation with three segments is modeled by a resistor connected to two resistors in parallel and the total resistance is given by \(R_p+\frac{(R_{d_1} R_{d_2})}{(R_{d_1}+R_{d_2})}\), where \(R_p\), \(R_{d_1}\) and \(R_{d_2}\) are resistance for the parent and two child segments, respectively. For each segment, the resistance is given by Poiseuille's law, \(R=\frac{8\mu l}{\pi r^4}\). Because the morphometric tree consists of bifurcation segments, the total resistance of a morphometric tree can be obtained by calculating the resistance backwards from the bottom elements to the root elements. Due to the Fahraeus Lindqvist effect, apparent viscosity decreases with decreasing diameter for vessels between 10 and 300 \(\upmu \hbox {m}\). An empirical formula for the viscosity was used (Pries et al. 1998; Secomb 2017). The apparent viscosity is given as follows:

where

and *D* is the diameter in micrometers, \(H_D\) is discharge hematocrit assuming \(H_D=0.45\), and \(\mu _p\) is the viscosity of plasma assuming \(\mu _p=1.24 \; cp\).

### Precapillary pressure estimation

The pressure tracing after occlusion of a distal PA initially decays rapidly and then attenuates slowly. The slow component represents the discharge of blood stored in the pulmonary microvasculature. The precapillary pressure is defined as the transition point between the rapid and slow components (Holloway et al. 1983). To estimate precapillary pressure, Siegel et al. (1989) used a circuit model-based biexponential function,

where \(\lambda \), \(\eta \), \(\phi \) and \(\gamma \) are parameters to be determined by curve fitting and \(P_\mathrm{la}\) is the left atrial pressure, to approximate the decay of PA pressure during wedging of the PA catheter. The precapillary pressure is given by

Pressure tracings from the balloon inflation to establishment of wedge pressure were obtained for each patient. Following Siegel et al. (1989), PA wedge pressure tracings were obtained and biexponential curve fitting was performed for each PAH patient. Figure 13 shows a PA wedge pressure tracing and a fitted biexponential function. Since catheterization was not available for controls, a representative normal PA tracing was used to get a precapillary pressure for all controls. Using the mean terminal pressure and flow of the 3D simulations, the resistance of a distal PA tree was derived and a morphometric tree was created to match that resistance. With the mean terminal flow of the 3D model, the WSS for each segment in the morphometric tree was calculated by using Poiseuille's law and variable apparent viscosity.

To check the sensitivity of distal WSS levels to the 3D outlet area, flow and pressure, we used a regression model (Kriging) to relate the 3D outlet pressure, flow and resistance to the outlet area. Then, the mean 3D outlet area was perturbed by a standard deviation and applied to the morphometric tree with the corresponding resistance and flow. For the 5 representative patients we tested, mean WSS levels in the distal tree were similar (Fig. 14).

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Yang, W., Dong, M., Rabinovitch, M. *et al.* Evolution of hemodynamic forces in the pulmonary tree with progressively worsening pulmonary arterial hypertension in pediatric patients.
*Biomech Model Mechanobiol* **18, **779–796 (2019). https://doi.org/10.1007/s10237-018-01114-0

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### Keywords

- Pulmonary arterial hypertension
- Wall shear stress
- Oscillatory shear index
- Wall strain
- Distal pulmonary artery
- Morphometry tree
- Blood flow simulation
- Patient-specific modeling