## 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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## References

Abman SH, Hansmann G, Archer SL, Ivy DD, Adatia I, Chung WKEA (2015) Pediatric pulmonary hypertension: guidelines from the American Heart Association and American Thoracic Society. Circulation 132(21):2037–2099. https://doi.org/10.1161/CIR.0000000000000329

Allen RP, Schelegle ES, Bennett SH (2014) Diverse forms of pulmonary hypertension remodel the arterial tree to a high shear phenotype. Am J Physiol Heart Circul Physiol 307(3):H405–H417

Barker AJ, Roldn-Alzate A, Entezari P, Shah SJ, Chesler NC, Wieben O, Markl M, Franois CJ (2015) 4d Flow assessment of pulmonary artery flow and wall shear stress in adult pulmonary arterial hypertension: results from two institutions. Magn Reson Med 73(5):1904–1913. https://doi.org/10.1002/mrm.25326

Barker AJ, Roldn-Alzate A, Entezari P, Shah SJ, Chesler NC, Wieben O, Markl M, Franois CJ (2015) Four-dimensional flow assessment of pulmonary artery flow and wall shear stress in adult pulmonary arterial hypertension: results from two institutions. Magn Reson Med 73(5):1904–1913. https://doi.org/10.1002/mrm.25326

Compton GL, Florence J, MacDonald C, Yoo SJ, Humpl T, Manson D (2015) Main pulmonary artery-to-ascending aorta diameter ratio in healthy children on MDCT. Am J Roentgenol 205(6):1322–1325. https://doi.org/10.2214/AJR.15.14301

Conover T (2007) Fluid diode simulation, pulmonary valve position: 2-d axisymmetric. Technical report unregulated right heart. Clemson University

Esmaily Moghadam M, Vignon-Clementel IE, Figliola R, Marsden AL (2013) A modular numerical method for implicit 0d/3d coupling in cardiovascular finite element simulations. J Comput Phys 244:63–79. https://doi.org/10.1016/j.jcp.2012.07.035

Esmaily-Moghadam M, Bazilevs Y, Marsden AL (2013) A new preconditioning technique for implicitly coupled multidomain simulations with applications to hemodynamics. Comput Mech 52(5):1141–1152. https://doi.org/10.1007/s00466-013-0868-1

Figueroa CA, Vignon-Clementel IE, Jansen KE, Hughes TJR, Taylor CA (2006) A coupled momentum method for modeling blood flow in three-dimensional deformable arteries. Comput Methods Appl Mech Eng 195(41–43):5685–5706

Ghorishi Z, Milstein JM, Poulain FR, Moon-Grady A, Tacy T, Bennett SH, Fineman JR, Eldridge MW (2007) Shear stress paradigm for perinatal fractal arterial network remodeling in lambs with pulmonary hypertension and increased pulmonary blood flow. Am J Physiol Heart Circul Physiol 292(6):H3006–H3018

Gleason RL, Humphrey JD (2004) A mixture model of arterial growth and remodeling in hypertension: altered muscle tone and tissue turnover. J Vasc Res 41(4):352–363. https://doi.org/10.1159/000080699

Holloway H, Perry M, Downey J, Parker J, Taylor A (1983) Estimation of effective pulmonary capillary pressure in intact lungs. J Appl Physiol 54(3):846–851

Hopper RK, Abman SH, Ivy DD (2016) Persistent challenges in pediatric pulmonary hypertension. Chest 150(1):226–236. https://doi.org/10.1016/j.chest.2016.01.007

Huang W, Yen RT, McLaurine M, Bledsoe G (1996) Morphometry of the human pulmonary vasculature. J Appl Physiol 81(5):2123–2133. https://doi.org/10.1152/jappl.1996.81.5.2123

Humphrey JD (2008) Mechanisms of arterial remodeling in hypertension: coupled roles of wall shear and intramural stress. Hypertension 52(2):195–200. https://doi.org/10.1161/HYPERTENSIONAHA.107.103440

Hunter KS, Lanning CJ, Chen SYJ, Zhang Y, Garg R, Ivy DD, Shandas R (2004) Computational fluid dynamic study of flow optimization in realistic models of the total cavopulmonary connections. Biomech Eng 128(4):564–72

Hunter KS, Albietz JA, Lee PF, Lanning CJ, Lammers SR, Hofmeister SH, Kao PH, Qi HJ, Stenmark KR, Shandas R (2010) In vivo measurement of proximal pulmonary artery elastic modulus in the neonatal calf model of pulmonary hypertension: development and ex vivo validation. J Appl Physiol 108(4):968–975

Ivy DD, Rosenzweig EB, Lemari JC, Brand M, Rosenberg D, Barst RJ (2010) Long-term outcomes in children with pulmonary arterial hypertension treated with bosentan in real-world clinical settings. Am J Cardiol 106(9):1332–1338. https://doi.org/10.1016/j.amjcard.2010.06.064

Jansen KE, Whiting CH, Hulbert GM (2000) A generalized-\(\alpha \) method for integrating the filtered navierstokes equations with a stabilized finite element method. Comput Methods Appl Mech Eng 190(3–4):305–319

Jeffery TK, Morrell NW (2002) Molecular and cellular basis of pulmonary vascular remodeling in pulmonary hypertension. Prog Cardiovasc Dis 45(3):173–202. https://doi.org/10.1053/pcad.2002.130041

Kassab GS, Rider CA, Tang NJ, Fung YC (1993) Morphometry of pig coronary arterial trees. Am J Physiol 265(1 Pt 2):H350–365. https://doi.org/10.1152/ajpheart.1993.265.1.H350

Kheyfets VO, O’Dell W, Smith T, Reilly JJ, Finol EA (2013) Considerations for numerical modeling of the pulmonary circulation—a review with a focus on pulmonary hypertension. J Biomech Eng 135(6):061,011. https://doi.org/10.1115/1.4024141

Kheyfets V, Thirugnanasambandam M, Rios L, Evans D, Smith T, Schroeder T, Mueller J, Murali S, Lasorda D, Spotti J, Finol E (2015a) The role of wall shear stress in the assessment of right ventricle hydraulic workload. Pulm Circ 5(1):90–100. https://doi.org/10.1086/679703

Kheyfets VO, Rios L, Smith T, Schroeder T, Mueller J, Murali S, Lasorda D, Zikos A, Spotti J, Reilly JJ, Finol EA (2015b) Patient-specific computational modeling of blood flow in the pulmonary arterial circulation. Comput Methods Programs Biomed 120(2):88–101. https://doi.org/10.1016/j.cmpb.2015.04.005

Lan H, Updegrove A, Wilson NM, Maher GD, Shadden SC, Marsden AL (2018) A re-engineered software interface and workflow for the open-source simvascular cardiovascular modeling package. J Biomech Eng 140(2):024501. https://doi.org/10.1115/1.4038751

Li M, Scott DE, Shandas R, Stenmark KR, Tan W (2009) High pulsatility flow induces adhesion molecule and cytokine mRNA expression in distal pulmonary artery endothelial cells. Ann Biomed Eng 37(6):1082. https://doi.org/10.1007/s10439-009-9684-3

Long CC, Hsu MC, Bazilevs Y, Feinstein JA, Marsden AL (2012) Fluid–structure interaction simulations of the Fontan procedure using variable wall properties. Int J Numer Methods Biomed Eng 28:513–527

Marsden AL, Esmaily-Moghadam M (2015) Multiscale modeling of cardiovascular flows for clinical decision support. Appl Mech Rev 67(3):030,804. https://doi.org/10.1115/1.4029909

McLaughlin VV, Presberg KW, Doyle RL, Abman SH, McCrory DC, Fortin T, Ahearn G (2004) American college of chest physicians: prognosis of pulmonary arterial hypertension: ACCP evidence-based clinical practice guidelines. Chest 126(1 Suppl):78S–92S. https://doi.org/10.1378/chest.126.1_suppl.78S

Migliavacca F, Pennati G, Dubini G, Fumero R, Pietrabissa R, Urcelay G, Bove EL, Hsia TY, de Leval MR (2001) Modeling of the Norwood circulation: effects of shunt size, vascular resistances, and heart rate. Am J Physiol Heart Circ Physiol 280(5):H2076–2086

Miyazaki S, Itatani K, Furusawa T, Nishino T, Sugiyama M, Takehara Y, Yasukochi S (2017) Validation of numerical simulation methods in aortic arch using 4d flow mri. Heart Vessels 32(8):1032–1044

Moghadam ME, Bazilevs Y, Hsia TY, Vignon-Clementel IE, Marsden AL, (mocha) MoCHA (2011) A comparison of outlet boundary treatments for prevention of backflow divergence with relevance to blood flow simulations. Comput Mech 48(3):277–291. https://doi.org/10.1007/s00466-011-0599-0

Mynard JP, Davidson MR, Penny DJ, Smolich JJ (2012) A simple, versatile valve model for use in lumped parameter and one-dimensional cardiovascular models. Int J Numer Methods Biomed Eng 28(6–7):626–641. https://doi.org/10.1002/cnm.1466

Odagiri K, Inui N, Hakamata A, Inoue Y, Suda T, Takehara Y, Sakahara H, Sugiyama M, Alley MT, Wakayama T, Watanabe H (2016) Non-invasive evaluation of pulmonary arterial blood flow and wall shear stress in pulmonary arterial hypertension with 3d phase contrast magnetic resonance imaging. Springerplus 5(1):1071. https://doi.org/10.1186/s40064-016-2755-7

Olufsen MS (1999) Structured tree outflow condition for blood flow in larger systemic arteries. Am J Physiol 276(1 Pt 2):H257–268

Pennati G, Migliavacca F, Dubini G, Pietrabissa R, de Leval MR (1997) A mathematical model of circulation in the presence of the bidirectional cavopulmonary anastomosis in children with a univentricular heart. Med Eng Phys 19(3):223–234

Pirat B, McCulloch ML, Zoghbi WA (2006) Evaluation of global and regional right ventricular systolic function in patients with pulmonary hypertension using a novel speckle tracking method. Am J Cardiol 98(5):699–704. https://doi.org/10.1016/j.amjcard.2006.03.056

Postles A, Clark AR, Tawhai MH (2014) Dynamic blood flow and wall shear stress in pulmonary hypertensive disease. Conf Proc IEEE Eng Med Biol Soc 2014:5671–5674. https://doi.org/10.1109/EMBC.2014.6944914

Potters WV, Ooij P, Marquering H, vanBavel E, Nederveen AJ (2015) Volumetric arterial wall shear stress calculation based on cine phase contrast mri. J Magn Reson Imaging 41(2):505–516

Pries AR, Secomb TW, Gaehtgens P (1998) Structural adaptation and stability of microvascular networks: theory and simulations. Am J Physiol 275(2 Pt 2):H349–360

Rol N, Timmer EM, Faes TJC, Vonk Noordegraaf A, Grnberg K, Bogaard HJ, Westerhof N (2017) Vascular narrowing in pulmonary arterial hypertension is heterogeneous: rethinking resistance. Physiol Rep 5(6):e13159. https://doi.org/10.14814/phy2.13159

Sahni O, Jansen KE, Shephard MS, Taylor CA, Beall MW (2008) Adaptive boundary layer meshing for viscous flow simulations. Eng Comput 24:267

Sakao S, Taraseviciene-Stewart L, Lee JD, Wood K, Cool CD, Voelkel NF (2005) Initial apoptosis is followed by increased proliferation of apoptosis-resistant endothelial cells. FASEB J 19(9):1178–1180. https://doi.org/10.1096/fj.04-3261fje

Sanz J, Kariisa M, Dellegrottaglie S, Prat-Gonzlez S, Garcia MJ, Fuster V, Rajagopalan S (2009) Evaluation of pulmonary artery stiffness in pulmonary hypertension with cardiac magnetic resonance. JACC Cardiovasc Imaging 2(3):286–295. https://doi.org/10.1016/j.jcmg.2008.08.007

Schafer M, Kheyfets VO, Schroeder JD, Dunning J, Shandas R, Buckner JK, Browning J, Hertzberg J, Hunter KS, Fenster BE (2016) Main pulmonary arterial wall shear stress correlates with invasive hemodynamics and stiffness in pulmonary hypertension. Pulm Circ 6(1):37–45

Schfer M, Ivy DD, Barker AJ, Kheyfets V, Shandas R, Abman SH, Hunter KS, Truong U (2017) Characterization of CMR-derived haemodynamic data in children with pulmonary arterial hypertension. Eur Heart J Cardiovasc Imaging 18(4):424–431. https://doi.org/10.1093/ehjci/jew152

Schmidt JP, Delp SL, Sherman MA, Taylor CA, Pande VS, Altman RB (2008) The simbios national center: systems biology in motion. Proc IEEE Specl Issue Comput Syst Biol 96(8):1266–1280

Secomb TW (2017) Blood flow in the microcirculation. Annu Rev Fluid Mech 49:443–461

Siegel LC, Pearl RG, Shafer SL, Ream AK, Prielipp RC (1989) The longitudinal distribution of pulmonary vascular resistance during unilateral hypoxia. Anesthesiology 70(3):527–532

Simonneau G, Gatzoulis MA, Adatia I, Celermajer D, Denton C, Ghofrani A, Sanchez MAG, Kumar RK, Landzberg M, Machado RF, Olschewski H, Robbins IM, Souza R (2013) Updated clinical classification of pulmonary hypertension. J Am Coll Cardiol 62(25, Supplement):D34–D41

Spilker RL, Feinstein JA, Parker DW, Reddy VM, Taylor CA (2007) Morphometry-based impedance boundary conditions for patient-specific modeling of blood flow in pulmonary arteries. Ann Biomed Eng 35(4):546–549

Szulcek R, Happ CM, Rol N, Fontijn RD, Dickhoff C, Hartemink KJ, Grnberg K, Tu L, Timens W, Nossent GD, Paul MA, Leyen TA, Horrevoets AJ, de Man FS, Guignabert C, Yu PB, Vonk-Noordegraaf A, Amerongen GPVN, Bogaard HJ (2016) Delayed microvascular shear adaptation in pulmonary arterial hypertension role. of platelet endothelial cell adhesion molecule-1 cleavage. Am J Respirat Crit Care Med 193(12):1410–1420

Tang BT, Pickard SS, Chan FP, Tsao PS, Taylor CA, Feinstein JA (2012) Wall shear stress is decreased in the pulmonary arteries of patients with pulmonary arterial hypertension: An image-based, computational fluid dynamics study. Pulm Circ 2(4):470–476. https://doi.org/10.4103/2045-8932.105035

Taylor CA, Hughes TJR, Zarins CK (1998) Finite element modeling of blood flow in arteries. Comput Methods Appl Mech Eng 158(1–2):155–196

Terada M, Takehara Y, Isoda H, Uto T, Matsunaga M, Alley M (2016) Low WSS and high OSI measured by 3d cine PC MRI reflect high pulmonary artery pressures in suspected secondary pulmonary arterial hypertension. Magn Reson Med Sci 15(2):193–202. https://doi.org/10.2463/mrms.mp.2015-0038

Tian L, Kellihan HB, Henningsen J, Bellofiore A, Forouzan O, Roldn-Alzate A, Consigny DW, Gunderson M, Dailey SH, Francois CJ, Chesler NC (2014) Pulmonary artery relative area change is inversely related to ex vivo measured arterial elastic modulus in the canine model of acute pulmonary embolization. J Biomech 47(12):2904–2910

Towns J, Cockerill T, Dahan M, Foster I, Gaither K, Grimshaw A, Hazlewood V, Lathrop S, Lifka D, Peterson GD, Roskies R, Scott JR, Wilkins-Diehr N (2014) XSEDE: accelerating scientific discovery. Comput Sci Eng 16(5):62–74. https://doi.org/10.1109/MCSE.2014.80

Truong U, Fonseca B, Dunning J, Burgett S, Lanning C, Ivy DD, Shandas R, Hunter K, Barker AJ (2013) Wall shear stress measured by phase contrast cardiovascular magnetic resonance in children and adolescents with pulmonary arterial hypertension. J Cardiovasc Magn Reson 15(1):81. https://doi.org/10.1186/1532-429X-15-81

Tuder RM, Marecki JC, Richter A, Fijalkowska I, Flores S (2007) Pathology of pulmonary hypertension. Clin Chest Med 28(1):23-vii. https://doi.org/10.1016/j.ccm.2006.11.010

Updegrove A, Wilson NM, Merkow J, Lan H, Marsden AL, Shadden SC (2017) SimVascular: an open source pipeline for cardiovascular simulation. Ann Biomed Eng 45(3):525–541. https://doi.org/10.1007/s10439-016-1762-8

Vignon-Clementel IE, Figueroa CA, Jansen KE, Taylor CA (2006) Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries. Comput Methods Appl Mech Eng 195(29–32):3776–3796

Vignon-Clementel IE, Figueroa CA, Jansen KE, Taylor CA (2010) Outflow boundary conditions for 3d simulations of non-periodic blood flow and pressure fields in deformable arteries. Comput Methods Biomech Biomed Eng 13(5):625–640

Yang W, Feinstein JA, Vignon-Clementel IE (2016) Adaptive outflow boundary conditions improve post-operative predictions after repair of peripheral pulmonary artery stenosis. Biomech Model Mechanobiol 15:1345–1353. https://doi.org/10.1007/s10237-016-0766-5

Yang W, Marsden AL, Ogawa MT, Sakarovitch C, Hall KK, Rabinovitch M, Feinstein JA (2018) Right ventricular stroke work correlates with outcomes in pediatric pulmonary arterial hypertension. Pulm Circ 8(3):2045894018780,534. https://doi.org/10.1177/2045894018780534

## 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

### 1.1 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.

### 1.2 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).

### 1.3 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 \(\mu _p=1.24 \; cp\).

### 1.4 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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DOI: https://doi.org/10.1007/s10237-018-01114-0