This work represents the first application of a microstructural orthotropic hyperelastic model to the clinical study of PA mechanics in children with PAH. A finite-element approach was adopted to implement the model. Material parameters determined from our previous studies via comparing model output to measured pressure-stretch results from normotensive and hypertensive MPA, LPA, and RPA obtained from a rat model of PAH [13
] were used in the study of PA mechanics in children. The long-term goal of this study is to understand the relationship between fundamental arterial microstructural changes and development of PAH and to use this information to develop clinically meaningful yet fundamentally based diagnostics for evaluating pulmonary vascular function noninvasively. This should be useful in following disease progression, especially regarding the initiation and effects of arterial remodeling in PAH, and in more accurately estimating pulmonary vascular reactivity in patients undergoing pulmonary vasodilator challenge as is routinely done at our institution [10
In our previous study, the material parameters in the microstructural model were determined based on biomechanical test on the medial layer of the PAs of rats with and without PAH [13
]. For the medial layer consisting of elastin-based laminae surrounded by extra-cellular matrix and smooth muscle cells, the microstructure can be represented as a tangled molecular network with individual molecular fibers connected to each other via cross-linking and entanglement. Although the eight-chain network model does not represent the actual organization/alignment of the protein fibers in the arterial wall, the molecular chains in the model represent the functional equivalent in a mechanical sense of the fiber network found in the medial layer. We have shown that the mechanical response of the medial layer is captured well by the entropic elasticity of a network of molecular chains. More importantly, because the material parameters in the microstructural model possess physical meanings, our previous study provoked the hypotheses that increase of cross-linking density might be a key mechanism by which pulmonary artery stiffens in PAH. Interested readers are referred to Zhang et al. [13
] for detailed discussions. Briefly, our previous results suggest that compared to normotensive conditions, hypertensive PAs showed no significant change in the chain density per unit volume (material parameter n
in ); however, chain extensibility decreased significantly (material parameter N
in ). The decrease in chain extensibility indicates that the mean spacing between molecular links has decreased in the hypertensive case, and implies greater degree of molecular cross-linking within, for example, elastin fibers to both other elastin fibers and/or to other structural protein fibers or smooth muscle cells found in the medial extra-cellular matrix. Although our efforts have focused initially on the medial layer of the arterial wall, it is important to recognize that the arterial wall is a complex multilayered structure and other components will contribute to its mechanical behavior as well. Subsequent models are being developed to allow inclusion of collagen fibers and smooth muscle cells and thereby expand the mechanical description of the artery wall.
4.1 Validation of the Model
Currently, the ability to obtain detailed spatially and temporally resolved clinical data are limited. Several diagnostics are performed on these patients; we routinely measure mean hemodynamics (PVR, cardiac output, mean pressures), vascular input impedance and pressure–diameter curves in the pulmonary vasculature for all patient studies in the catheterization laboratory [10
]. The P-D
response of the RPA is an easily implemented method of evaluating structural response of the upstream arteries under vasodilator challenge. The P-D
response was thus used in this work to validate the finite element modeling (). We use a simple tube model for the validation, since our primary interest here is on the pressure–diameter response rather than detailed stress/strain distribution. In addition, the use of clinical P-D
data, albeit only for the RPA, provides an initial approach to the issue of validation for complex vascular structures under in vivo conditions. The orientations of the clinical P-D
loops of normotensive and hypertensive arteries agree reasonably with the P-D
responses using material parameters based on animal studies [13
]. The hypertensive P-D
loop has higher mean PA pressure and steeper loop orientation, and the hypertensive P-D
loop shifts to the left of the normotensive loop. This indicates that passive changes, i.e., movement of the P-D
loop along the nonlinear stress–strain curve as pressure varies [24
] is not the only mechanism involved in clinical manifestation of stiffened pulmonary arteries. Other mechanisms such as structural remodeling and active myogenic effects (increase in smooth muscle tone) may also play important roles. Our previous work provoked the hypothesis that cross-link density is one mechanism by which structural remodeling may take place in the hypertensive pulmonary artery [13
]. Results presented here appear to further confirm this hypothesis since varying molecular cross-linking density in the model allows us to simulate the changes in the P-D
loops between normotensive and hypertensive conditions reasonably well, as shown in . However, active mechanisms such as smooth muscle relaxation and/or constriction can also change the intrinsic elastic properties of the PAs, and thereby produce a shift of the P-D
]. It is likely a combination of such effects that dictate the ultimate structural characteristics of these arteries. Combining clinical data extraction of the P-D
loop under baseline and challenge conditions with finite-element modeling studies should thus provide heretofore unavailable clues regarding structural remodeling changes of the pulmonary artery in the clinical manifestation of pulmonary hypertension.
4.2 Effect of Initial Conditions: Axial Stretch
The axial stretch affects the P-D response of the artery, as well as the stresses in the arterial wall (Figs. and ). Basically, as the axial stretch increases, the artery becomes stiffer in the circumferential direction, which is manifested by the P-D curves. Stresses increase in both the circumferential and longitudinal directions. Movement of the heart may possibly cause stretch/twist of the pulmonary vasculature before or during inflation; this effect is not included in this study but is a topic of future research.
4.3 Effect of Initial Conditions: Material Orthotropy
Although in the validation of the model we used averaged material parameters, (n
), and assumed the arterial wall to be isotropic (), we did observe obvious anisotropic response in the longitudinal and circumferential directions of many proximal PAs in our previous animal study [13
]. Orthotropy of the artery has also been demonstrated and studied both experimentally and theoretically on animals by other research groups [21
]. Study of bovine carotid arteries by von Maltzahn [21
] suggests that both the medial and adventitial layers are stiffer in the axial direction than in the tangential direction. Zhou and Fung’s study [27
] on thoracic aortas of dogs indicates that the longitudinal stress–strain curve turns nonlinear at a smaller strain and has as a steeper nonlinear region than the circumferential stress–strain curve does. Studies of pulmonary arteries of rats with PAH by Drexler et al. [28
] show that most of the MPAs are stiffer in the circumferential direction and almost all of the LPAs and RPAs are stiffer in the longitudinal direction. In this study, we assume that the PAs have the same material properties in a 3D vascular system. However it is possible in one patient that the PAs may have a different orthotropy trend. Our validation based on P-D
responses at the RPA cannot resolve this complex issue. Further work using more advanced imaging techniques such as magnetic resonance tissue tagging to examine arterial strain patterns should shed further light on this topic.
To study the effect of orthotropy on 3D pulmonary vascular mechanics, the chain density (n
) and chain length (N
, number of rigid links within each chain) is kept the same, and the normalized unit element dimensions (a
, and c
) are varied, which at the microlevel results in changing the spatial orientation of the macromolecular chains in the arterial wall, and at the macrolevel results in changing the material property, i.e., cylindrical orthotropy. Since a
are aligned with the circumferential and longitudinal direction of the arterial wall, the angle ArcTan[b
] reflects the projection of the macromolecular chains onto the arterial wall. This perspective is similar to Holzapfel’s constitutive model [29
], in which each layer is considered as a composite reinforced by two families of fibers that are arranged in symmetrical spirals. In their model, fiber angle is defined as the angle between the fibers and the circumferential direction of the artery. The fiber angles in Figs. – correspond to 45 deg, 60.3 deg, and 29.7 deg. In Holzapfel’s multilayer consitutive model of the arterial wall, this angle was assumed to be 29 deg in the media and 62 deg in the adventitia, based on the work from Xie et al. [23
Considering the artery wall as a cylindrically orthotropic material results in different stress/strain patterns within the 3D vasculature, as shown in Figs. –. Different P-D
responses can result purely due to the orthotropy in the arterial wall, as shown in . Many three-dimensional finite-element simulations of arteries [19
] have focused more on residual stresses and assumed the artery to be isotropic. Whether anisotropy should be included in such models is an important question, especially given the increased complexity this adds to the model, Our studies show marked differences in stresses and strains within the branched vasculature between isotropic and orthotropic conditions, which might be important, since physiologically fibroblasts, smooth muscle cells and a variety of precursor cells may alter function based on local material stresses and strains [1
]. Thus, further studies on linking the change in artery mechanics to remodeling, and obtaining further clinical data to more completely investigate this issue are necessary.
Note that the boundary conditions of applying movement planes at the ends allow the deformed shape of the 3D vasculature to differ for different anisotropy assumptions. For example in , more longitudinal elongation is presented in than in , and longitudinal shortening is seen in . When the material is assumed to be orthotropic and stiffer in the longitudinal direction (HOL), it is difficult for the vasculature to be stretched in the longitudinal direction. Therefore when the vasculature is inflated, the diameter of the artery increases and the ends expands in the circumferential direction. The combination of the constriction of the end movement plane and expansion of the PA produces negative longitudinal strain as shown in .
It is also useful to address the assumption of local wall thickness being 10% of local diameter. Our biplane angiography method does not provide information on wall thickness. We are examining other methods such as 3D MRI, which can provide such information; however this work is still in its early stages. At the pressure of 26 mm Hg, increasing the wall thickness by 60% results in a change of about 6% in the wall diameter (by comparing HI and HI (16%) in ), which is comparable to the total deformation, about 20% of the initial diameter. It is interesting to note that changing material properties from isotropic to orthotropic produces more variation in the results than changing wall thickness.
4.4 Limitations in Our Study
It is important to recognize the limitations in our study. Prediction of patient-specific pulmonary artery mechanics is based on a microstructural constitutive model developed from testing of rat pulmonary arteries. Only material properties of the intimal-medial layer are considered in the finite element model, which may underestimate the stiffness of the PA. Modeling of the 3D pulmonary vasculature involves many complicated issues such as artery geometry, appropriate boundary and loading conditions, residual stresses, and unknown material properties. Given the absence of detailed information on patient-specific arterial wall material properties, cross-sectional geometry of the patient’s vascular system, etc., available using current clinical diagnostics, we assume the initial cross section of the artery to be circular, and the wall thickness to be 10% of the local diameter. Our current efforts using magnetic resonance imaging (MRI) to resolve the details of arterial wall geometry in patients should provide a means to verify and/or refine these assumptions.
Uniform pressure was used and pressure drop in the vascular tree is not considered. Movement of the heart, which may produce deformation, such as stretching and twisting, of the PAs is currently under study and not included in this work. For the orthotropic assumption, the alignment of the local material directions in the artery wall, especially at the junctions of branches, is unknown and its effect on the local stress/strain field needs to be investigated further. Only MPA, LPA, and RPA are included in this study and any effects due to higher order branching systems are not considered. Interpatient variability is an important issue and requires significant effort in obtaining precise data in infants and small children. We have put such efforts into place and are collecting data but this is a slow process that will take several months to obtain a reasonable number of patient data points.
Residual stress was not included in this work. Effect of residual stress in blood vessels has been studied extensively in the literature both experimentally and theoretically [30
], mainly for the purpose of understanding remodeling in the blood vessel walls and the ability of the vessel to adapt to mechanical loading. A common approach of including residual stress in cylindrical models is to form the cylinder (load-free condition) by closing an open sector (stress-free condition) [31
]. The open angle of the sector is usually determined experimentally by cutting an unloaded ring segment radially and measuring the angle subtended between two lines originating from the midpoint of the inner wall to the tips of the inner wall [30
]. Such studies have consistently shown that accounting for residual stress in the stress analysis of arteries under physiologic pressure substantially reduces the stress gradient in the circumferential and axial stress across the arterial wall. Using a similar approach, Delfino et al. [19
] included residual stress in a 3D finite-element model of carotid artery bifurcation based on cutting cadaver specimens along different lines and observing their “opening” behavior. However this approach is difficult to realize when clinical data and/or complex 3D anatomy are involved.
In conclusion, this work represents the first application of a statistical mechanics based orthotropic hyperelastic model to the clinical study of PA mechanics. Finite-element modeling using material properties from our previous results of rats with PAH agrees reasonably with the clinical P-D response in children with PAH. Material orthotropy affects the stress/strain pattern and P-D response. Further studies including higher order branching, non-uniform pressure loading, addition of the adventitial layer as well as active elements simulating smooth muscle cell action and use of 3D images derived from MRI are underway.