The complex distribution of arterial system mechanical properties, pulsatile hemodynamics, and vascular stresses emerges from three simple adaptive rules
Abstract
Arterial mechanical properties, pulsatile hemodynamic variables, and mechanical vascular stresses vary significantly throughout the systemic arterial system. Although the fundamental principles governing pulsatile hemodynamics in elastic arteries are widely accepted, a set of rules governing stress-induced adaptation of mechanical properties can only be indirectly inferred from experimental studies. Previously reported mathematical models have assumed mechanical properties adapt to achieve an assumed target stress “set point.” Simultaneous prediction of the mechanical properties, hemodynamics, and stresses, however, requires that equilibrium stresses are not assumed a priori. Therefore, the purpose of this work was to use a “balance point” approach to identify the simplest set of universal adaptation rules that simultaneously predict observed mechanical properties, hemodynamics, and stresses throughout the human systemic arterial system. First, we employed a classical systemic arterial system model with 121 arterial segments and removed all parameter values except vessel lengths and peripheral resistances. We then assumed vessel radii increase with endothelial shear stress, wall thicknesses increase with circumferential wall stress, and material stiffnesses decrease with circumferential wall stress. Parameters characterizing adaptive responses were assumed to be identical in all arterial segments. Iteratively predicting local mechanical properties, hemodynamics, and stresses reproduced five trends observed when traversing away from the aortic root towards the periphery: decrease in lumen radii, wall thicknesses, and pulsatile flows and increase in wall stiffnesses and pulsatile pressures. The extraordinary complexity of the systemic arterial system can thus arise from independent adaptation of vessels to local stresses characterized by three simple adaptive rules.
although there is a high degree of variation in arterial mechanical properties, pulsatile hemodynamics, and vascular stresses throughout the systemic arterial system, investigators have identified consistent trends when traversing from the aortic root towards the peripheral arteries. Attending a decrease in vessel lumen radius are a decrease in wall thickness and an increase in wall stiffness (24). As expected in a ramifying arterial tree, mean and pulsatile blood flow decrease with each branch encountered. In contrast, attending an expected small decrease in mean transmural pressure is a less intuitive increase in pulse pressure (systolic-diastolic pressures) toward the periphery (45). Although arterial stresses are rarely measured in vivo, circumferential wall stress and endothelial shear stress tend to decrease (2–4, 19, 33, 34). Explanations of the origin of these trends have necessarily been incomplete, because they either relied on teleological arguments (16, 59, 61) or focused on a subset of complexities by assuming heterogeneous arterial structures (32, 47, 55, 60, 62), hemodynamic properties (16, 28, 59, 61), or mechanical stresses (8, 17, 23, 41, 43, 57, 58) a priori. It is generally agreed, however, that the interactions among network hemodynamics, local stresses, and adaptation of mechanical properties give rise to the observed complexities of the mammalian arterial system (13, 38).
Although the principles governing pulsatile blood pressures and flows in the human systemic arterial system are widely accepted (32, 47, 55, 60, 62), a set of rules governing structural adaptation can only be indirectly inferred from experimental studies. Investigators have long reported that chronic changes in blood pressure and flow in vivo are associated altered mechanical properties of vessels (48, 63). Modern mechanobiology has revealed that it is not pressure and flow that directly stimulate adaptive responses in the large elastic arteries but rather the mechanical stresses acting on the arterial wall (13, 49). The inability to rigorously control stresses in vivo arises from a complex interaction among arterial mechanical properties, hemodynamic variables, and vascular stresses. Changes in stresses affect mechanical properties, changes in mechanical properties in turn affect hemodynamics, and changes in hemodynamics in turn affect vascular stresses. Despite the difficulty in ascribing causality from correlation of one particular vascular stress and one particular mechanical property, investigators believe that endothelial shear stress is the predominant determinant of vessel radius (13) and circumferential wall stress is the predominant determinant of wall thickness and material stiffness (9, 24).
Given the difficulty in experimentally establishing a causal relationship between changes in a particular mechanical property to a particular stress in vivo, investigators have relied heavily on mathematical modeling to characterize the structural adaptation of single vessels (8, 17, 57, 58) or entire vascular networks (41, 44). Such models have not only confirmed the critical role played by endothelial shear stress and circumferential wall stress, they yielded the critical insight that a complex architecture can emerge from the identical adaptive rules universally applied to each vessel. Perhaps given the complexity arising from pulse wave propagation and reflection (30, 46, 60) in arterial networks, all but one recent model (23) of an adapting arterial network have focused on nonpulsatile hemodynamics, and none have addressed adaptation of arterial stiffness. Simultaneous prediction of the observed heterogeneous mechanical properties, pulsatile hemodynamics, and vascular stresses requires a set of adaptive rules be defined without assuming equilibrium a priori. Motivated by the observation that adaptation reduces the initial perturbation in vascular stresses in vivo (13), mathematical models purporting to explain the structural adaptation of vascular networks have universally assumed vascular adaptation to target stress “set points” (8, 17, 23, 43, 44, 57, 58). Cecchini et al. (5), however, questioned whether the set point is a distinct structural entity in biological control. Their seminal work illustrated that the set point concept, borrowed from engineered systems, is unnecessary to explain control of physiological systems. Instead, homeostasis can be achieved when a system with competing processes (providing negative feedback) yields an equilibrium “balance point.” Therefore, the purpose of this work is to use a balance point approach to identify the simplest set of universal adaptation rules that simultaneously predict observed mechanical properties, pulsatile hemodynamics, and vascular stresses throughout the human systemic arterial system.
METHODS
Pulsatile pressure and flow in a realistic human arterial network.
To characterize the relationship between blood pressure and flow in an arterial network, we relied on a standard one-dimensional, “transmission line” approach typically used for characterizing pulsatile hemodynamics in large arterial networks, described in detail elsewhere (24, 30). Briefly, only the primary effects of blood mass, viscosity, vessel geometries, and vessel wall mechanical properties are considered. The secondary effects of rotational flow, radial flow, body forces, and entrance phenomena are assumed to be negligible (30). From these assumptions, the standard Navier-Stokes equation may be simplified into a first-order partial differential equation relating the change in blood pressure (P) to flow (Q) as a function of time (t) and axial position. Noordergraaf (30) and Westerhof et al. (60) provided a detailed and careful analysis to show that the resulting partial differential equation can then be approximated by an ordinary differential equation for a segment of an artery if its length (Δz) is sufficiently short.
Assumed vascular network structure.
We implemented the classical Westerhof model (60), which applies Eqs. 1–3 to a distributed human systemic arterial network model consisting of 121 arterial segments (Fig. 1). The details of the model are described elsewhere (31, 60), as well as validation in later implementations (47, 55). Following our previous approaches (25, 26, 42), the coronary circulation was omitted in the present work. The original values for arterial segment lengths and terminal resistances reported by Westerhof et al. (60) were employed for the present work. We retained the segmentation (Δz) of arteries employed by Westerhof et al. (60) not only to speed computation but also to provide a discrete value of r, h, and E that can adapt to a single value of stress within a particular segment.
Fig. 1.Illustration of the systemic arterial network model consisting of 121 vessel segments drawn to scale according to radii and lengths reported by Westerhof et al. (60). Arteries are terminated by resistances with values also reported by Westerhof et al. (60).
Assumed input aortic flow.
To simplify, we assumed an input blood flow from the heart into the aortic root. The inlet pulsatile flow was reproduced from the waveform reported by Stergiopulos et al. (54). The magnitude and period were adjusted to produce a systolic pressure of 120 mmHg and a diastolic pressure of 80 mmHg in the ascending aorta of the modified Westerhof model (25, 26, 42, 60) at heart rate of 72 beats/min.
Mechanical stresses.
To calculate the average circumferential wall stress (i.e., the stress averaged in the radial direction from intima to adventitia), we used Laplace's Law applied to a thick-walled vessel (9, 12). The time-dependent circumferential wall stress (σ) was calculated using the time-dependent pressure at the entrance of the vessel.
Modeling adaptive responses of arteries to mechanical stress.
To simplify, we assumed that radius (r), wall thickness (h), and elastic modulus (E) adapt independently of each other in response to vascular stresses. Then, we assumed that radius adapts in response to shear stress while wall thickness and elastic modulus adapt in response to wall stress. These rules were based on consensus for the most commonly assumed stimulus for each mechanical property (8, 15, 17, 38, 41, 43, 57, 58). Two principles further constrained our formulation of the adaptive rules. First, a complex, nonlinear function may be approximated by a linear one. Second, adaptive processes must provide negative feedback to ensure stability. Taken together, these assumptions lead to three simple linear equations.
Iterative method to determine equilibrium variables.
Equations 1–8 form a system of equations that can be iteratively solved for the variables r, h, E, P, Q, σrms, and τrms assuming parameters characterizing adaptation (ro, α, ho, β, Eo, and γ) have the same values for all vessel segments in the network. The iterative process to calculate equilibrium variables can be summarized: 1) assume initial values for r, h, and E for all segments; 2) calculate values of P(t) and Q(t) for each segment using Eqs. 1–3; 3) calculate σrms, and τrms using Eqs. 4–7; and 4) calculate new values of r, h, and E from σrms and τrms using Eq. 8. Steps 2–4, illustrated in Fig. 2, are iterated until steady state is reached. We assumed steady state was adequately achieved when the values of r, h, and E were within 0.001% of the values obtained in the previous iteration. Throughout the simulation, the parameters for adaptive rules (ro, α, ho, β, Eo, and γ) were kept constant. The initial values of r, h, and E for all vessel segments were arbitrarily chosen to be 1 mm, 1 mm, and 100 kPa, respectively. To verify that equilibrium was insensitive to initial conditions, we performed two additional simulations with different initial conditions: randomly generated positive values of r, h, and E; and σrms and τrms with values of 1 Pa. To reduce error in pressure pulse morphology, all arterial mechanical properties were allowed to adapt iteratively except the radii of the ascending aorta, which were kept constant at values originally reported by Westerhof et al. (60).
Fig. 2.Iterative process to calculate equilibrium hemodynamic variables, vascular stresses, and mechanical properties. From initial values for radius (r), wall thickness (h), and elastic modulus (E), the values of pressures (P) and flows (Q) were calculated for vessel segments illustrated in Fig. 1. From pressures and flows, mechanical stresses (σrms, τrms) were calculated for each vessel. Finally, new values for r, h, and E were calculated using adaptive rules. The process is repeated until steady state was achieved.
Procedure to estimate values of parameters characterizing the adaptive process.
To simultaneously predict mechanical properties, hemodynamic variables, and stresses, the parameters characterizing the adaptive process ro, α, ho, β, Eo, and γ must be prescribed a priori. Without the ability to rigorously control vascular stresses in vivo, these parameter values cannot be directly measured but instead must be inferred indirectly from in vivo data. First, we assumed the original values of r, h, and E reported by Westerhof et al. (60), with one major caveat. Because Stergiopulos et al. (55) illustrated that the vascular compliances of the original Westerhof model were ∼50% too low, to be consistent with subsequent models (25, 26, 42), we assumed values of E were 50% lower than those reported by Westerhof et al. (60). The corrected Westerhof model has been subsequently validated (47, 55). We then estimated parameter values by linear regression from the relationships among r, h, and E and computed values of σrms and τrms, respectively. Once we confirmed that the inferred parameter values of ro, α, ho, β, Eo, and γ can yield a stable equilibria, they were then fine-tuned by curve-fitting procedures to better approximate values of r, h, E, as well as corresponding calculated pulse pressure and pulse flow (peak-trough). In particular, we used a simple gradient descent method to minimize a “cost function,” defined as average of the mean percentage errors in predicted r, h, E, pulse pressure, and pulse flow.
Graphical balance point analysis to characterize equilibrium structure and stress.
To illustrate the interaction between mechanical properties (r, h, and E) and vascular stresses (σrms and τrms), we constructed graphs indicating equilibrium balance points for the first vessel segment of the descending aorta. First, we solved the hemodynamic (Eqs. 1–3) and the endothelial shear stress equations (Eqs. 5 and 7) and graphed τrms as r was altered over a wide range. This relationship characterizes the segment's biomechanics. The structural variables of all other arterial segments were kept constant in the process. We then assumed Eq. 8a and graphed r as τrms was altered over a wide range. This relationship characterizes the segment's mechanobiology. The intersection of the two graphs represents a balance point yielding equilibrium values of r and τrms. Second, we solved the hemodynamic (Eqs. 1–3) and the circumferential wall stress equations (Eqs. 4 and 6) and graphed σrms as h was altered over a wide range. We then assumed Eq. 8b and graphed h as σrms was altered over a wide range. The intersection of the two graphs represents a balance point yielding equilibrium values of h and σrms. Third, we solved the hemodynamic (Eqs. 1–3) and the circumferential wall stress equations (Eqs. 4 and 6) and graphed σrms as E was altered over a wide range. We then assumed Eq. 8c and graphed E as σrms was altered over a wide range. The intersection of the two graphs represents a balance point yielding equilibrium values of E and σrms.
Graphical illustration of emergent structural heterogeneity from homogenous adaptive rules.
To illustrate the principle that a homogenous adaptive rule can yield heterogeneous mechanical properties, we constructed a standard balance point graph for four representative vessel segments along a pathway from the descending aorta to the femoral artery illustrated in Fig. 1. In this case, the relationships between the shear stresses and radii, representing biomechanics of each vessel, were solved from Eqs. 1–3, 5, and 7 and plotted. On the same graph, the universal adaptive rule (Eq. 8a) characterizing adaptation of radius to endothelial shear stress was plotted. Multiple intersections represent multiple, heterogeneous equilibria arising from the unique mechanical environment of each vessel segment.
Pressure morphology.
To illustrate changes in pressure pulse morphology along the aorta, we plotted transmural pressure versus time over a cardiac cycle for seven vessel segments at increasing distance from the aortic root. The distance from the aortic root was calculated by adding vessel segments lengths defined by Westerhof et al. (60). To compare with classical plots of pressure morphology (29), graphs were offset.
Prediction of arterial system mechanical properties compared with accepted values.
Predicted values of r, h, and E were plotted for each of the segment of the Westerhof model, along with the previously reported values (60).
Prediction of pulse pressures and flows.
Reported average values of in vivo measurements of pulse pressure and pulse flow were compiled and compared with model predictions.
RESULTS
Numerical values for adaptive parameters.
Values of parameters resulting in stable equilibria for r, h, and E were identified for the adaptive rules listed in Eq. 8 (Table 1). There were only five nonzero values, since the value of ho was 0.
| Parameters | Values |
|---|---|
| ro | 0.041 cm |
| α | 0.044 cm3/dyn |
| ho | 0 cm |
| β | 1.44 × 10−7 cm3/dyn |
| Eo | 1,500 kPa |
| γ | 1.27 × 10−5 |
Graphical representations of balance points.
The transfer function characterization of the interaction of biomechanics (solid box) and mechanobiology (dashed box) is illustrated in Fig. 3, left. As concluded by Cecchini et al. (5), there is no need for a particular a priori assumption of a set point. The graphical balance point characterization of the interaction of biomechanics (solid curves) and mechanobiology (dashed lines) is illustrated in Fig. 3, right. Equilibrium values of r, h, and E as well as σrms and τrms are represented by the intersection of the curves (circles). For each graph, biomechanics and mechanobiology curves have opposite slopes, indicating the presence of negative feedback.
Fig. 3.Transfer functions (left) and balance point graphs (right) characterizing interaction of biomechanics and mechanobiology for the first segment of the descending aorta. Solid lines represent vascular biomechanics, the effect of vascular mechanical properties (radius r, wall thickness h, and wall material stiffness E) on vascular stresses (endothelial shear stress τrms and circumferential wall stress σrms). Dashed lines represent vascular mechanobiology, the effect of vascular stresses on vascular mechanical properties. The intersections of the plots (circles) represent equilibria. A: interaction of radius r and endothelial shear stress τrms. B: interaction of wall thickness h and circumferential wall stress σrms. C: interaction of wall material stiffness E and circumferential wall stress σrms.
Graphical illustration of emergent heterogeneity.
Figure 4 illustrates the equilibrium balance points for radii and shear stresses for arterial segments along the aortic-femoral pathway (Fig. 1). The biomechanics curves (solid curves) are different, because arterial segments are exposed to different flows within the arterial network. For the same adaptive rule (dashed line) imposed on all vessels, unique balance points emerge, leading to heterogeneity in both vascular mechanical properties and vascular stresses.
Fig. 4.Graphical illustration of the interaction between local endothelial shear stress and vessel radius for 4 representative vessel segments illustrated in Fig. 1. The solid curves (biomechanics) represent how each vessel's radius affects its endothelial shear stress. These curves are different because they are exposed to different flows along different portions of the arterial network. The dashed curve (mechanobiology) represent a single adaptive response imposed on all vessels (Eq. 8a). Heterogeneity in radius and shear stress thus arises from a universal adaptive response. (DeA, descending aorta; AbA, abdominal aorta; IlA, iliac artery; FeA, femoral artery).
Prediction of stable equilibria.
Simulating the simultaneous adaptation of r to τrms, h to σrms, and E to σrms with parameters given in Table 1 resulted in stable equilibria for r, h, E, P, Q, and τrms and σrms. All values of r, h, and E increased from the prescribed initial values of 1 mm, 1 mm, and 100 kPa, respectively. Equilibria were not sensitive to initial values assumed. Alternative simulations with different initial conditions resulted in the same equilibrium values.
Predicted and reported values of equilibrium mechanical properties.
In equilibrium, all values for r, h, and E are unique to each vessel segment. Figure 5 compares the values of r, h, and E arising from the adaptive rules (solid bars) to the compared with values reported by Westerhof et al. (60), corrected by Stergiopulos et al. (55) (grey bars). Although there were notable differences in some of the peripheral vascular stiffnesses, similar patterns of heterogeneity were reproduced.
Fig. 5.Radii (A), wall thicknesses (B), and elastic moduli (C) predicted from the adaptive model compared with values reported by Westerhof et al. (60), corrected by Stergiopulos et al. (55). Vessel segment numbers are identical to those used by Westerhof et al. (60).
Pressure pulse morphology.
At increasing distances from the aortic root, systolic pressure gradually increases while diastolic pressure slightly decreases (Fig. 6). There is an increasing time shift with increasing distance from the aortic root. The morphology of the pressure waveform also changes.
Fig. 6.Predicted time-dependent blood pressure in vessel segments along the aortic-femoral pathway illustrated in Fig. 1. Distances (2 cm to 25 cm) in graph represent the distance of the pressure pulse from the aortic root. A 10-mmHg scale is placed in figure for reference. Consistent with reported trends (24, 29), the systolic pressure shows a gradual increase along the aortic-femoral pathway, while diastolic pressure slightly decreases.
Predicted and reported values of equilibrium pulse pressures and flows.
Predicted values of pulsatile pressures and flows for various arteries are compared with in vivo measurements reported for young, healthy human subjects in Fig. 7.
Fig. 7.Comparison between model predictions and reported values for pulse pressure (A) and pulse flow (B). Open circles indicate model results. Numbers next to reported data (*) denote the relevant reference: (10, 11, 14, 18, 20, 21, 27, 35, 36, 47, 50–52, 56).
DISCUSSION
The five most salient features of the systemic arterial circulation were predicted with three adaptation rules.
The present work has identified three simple adaptive rules (Eq. 8) that simultaneously predict the equilibrium mechanical properties, pulsatile hemodynamics, and vascular stresses throughout the human systemic arterial system. The five reported trends from the aorta to the femoral artery (Figs. 5–7) were reproduced without resorting to teleological arguments or a priori assumptions of equilibrium vascular stresses. In the process, the present work transformed the 360 local parameter values of radii, wall thicknesses, and elastic moduli employed by Westerhof et al. (60) into variables generated by as few as five nonzero adaptive parameters (Table 1). The complexity of the systemic arterial system emerged from accepted equations characterizing pulsatile blood pressure and flow (Eqs. 1–3), vascular stresses (Eqs. 4–7), and commonly assumed mechanical stimuli for vascular adaptation. Although the errors in each of the predicted values compared with reported values are not insignificant, they reproduce the same trends in mechanical properties (Fig. 5) and hemodynamics (Fig. 7). The present work thus predicts a significant amount of complexity with a model that is relatively simple.
Limitations arising from chosen scale and scope.
We adopted three guidelines to constrain the scale and scope of the present work. First, we focused on the interaction of only two hierarchical levels. Although the systemic arterial system interacts with the heart, veins, and the pulmonary circulation, we assumed constant input flow and terminal resistances. Furthermore, although elastic arterial segments have notable nonuniform mechanical properties (especially around curves and bifurcations), we further assumed each arterial segment was a uniform cylinder characterized by a single set of r, h, and E. This constraint was necessary to avoid invoking adaptive responses at the cellular scale to reproduce complicated geometries. Second, we limited the adaptive stimuli to local mechanical stresses to avoid invoking extrinsic control mechanisms such as hormonal and neural stimuli. Although some of the vessels included in the present work are muscular and are affected by nonmechanical regulatory mechanisms, we focused on very long time scales where mechanical stimuli are dominant (13). Third, we limited primary stimuli to circumferential wall stress and endothelial shear stress (Eq. 8) to avoid invoking secondary effects of longitudinal stretch and torsion. Although changes in length affect circumferential stress (12), we constrained the dimensional scale to the radial direction by assuming segmental lengths were constant. Taken together, strictly adhering to the chosen hierarchical levels in the present work not only avoided compounding complexity but also limited the number of assumptions necessary to produce meaningful results.
Determining values of adaptive parameters that cannot be measured empirically.
The present work addresses adaptive processes from a modeling perspective, because currently there are no means to experimentally control pulsatile stresses in an artery chronically. Because arteries are particularly difficult to culture in vitro, most mechanobiology studies are performed on cultured cells to control mechanical stresses (6, 13). Although radii and wall thicknesses can be measured in humans noninvasively, there is currently no means to continually adjust both pulsatile blood pressure and flow in vivo to control vascular stresses. Notwithstanding the experimental challenges of even estimating in vivo vascular stresses in vivo (2–4, 19, 33, 34, 53), there is a general consensus that the primary determinant of radius is endothelial shear stress (13, 40) and the primary determinant of wall thickness and stiffness is circumferential wall stress (9, 24). Although the assumed adaptive responses (Eq. 8) embody this consensus, parameter values characterizing adaptation as of yet cannot be measured directly. The present work thus used modeling to infer parameter values characterizing adaptive responses by fitting a very complex model to a complete set of structural data (Fig. 5). This approach was inspired by that of Pries and Secomb (38–41), who similarly used such inferences to make numerous fundamental contributions to microvascular physiology.
Limitations arising from assuming linear adaptive processes.
Besides limiting the number of unknown parameters by constraining the scope and scale of the model, we added two additional constraints to our formulation of adaptive responses. First, we assumed only one mechanical stimulus for each mechanical property to avoid invoking a minimum of three more parameters to characterize potential interactions (38, 39). Second, we assumed linear functions for the adaptive rules to avoid adding a minimum of six more parameters to characterize more realistic sigmoidal functions. This more complicated characterization would be particularly interesting, because when a stress (e.g., endothelial shear stress) is below a “threshold” or above a “saturation level,” the associated structural variable (e.g., radius) becomes relatively constant. In this case, the remaining adaptive responses yielded more extreme values for structural variables (e.g., wall thickness and material stiffness) to compensate. We had therefore considered relaxing these strict constraints. However, introducing additional parameters that could not be verified empirically would make it much easier for a model to reproduce the limited data available (1). In fact, our very conservative modeling approach embodied by Eq. 8 still required five unknown nonzero parameter values (ro, Eo, α, β, and γ), evoking a quote attributed to John von Neumann (7), “With four parameters I can fit an elephant. With five I can make his trunk wiggle.” The resulting simple first-order adaptive scheme nonetheless reproduced 360 structural parameters and hemodynamic trends observed when traversing away from the aortic root towards the periphery (Figs. 5 and 7). By valuing simplicity of assumptions over accuracy of results, we were able to identify the simplest set of adaptation rules that are sufficient to predict the complex distribution of the systemic arterial system mechanical properties, hemodynamic variables and stresses.
Limitations arising from first-order approximations for mechanical properties, vascular stresses, and pulsatile hemodynamics.
As discussed above, the present work reduced mechanical properties to a single length, radius, wall thickness, and material stiffness in uniform cylindrical segments to avoid the compounding complexity of modeling adaptive processes at a cellular level. The discrepancy between predicted and reported radii and wall thicknesses in the aortic root (Fig. 5) may in part arise from such geometric complexities that are merely approximated by first-order description of vascular geometry. Furthermore, we could have attempted to assume nonlinear elastic properties, if they were well quantified for each segment of the human systemic arterial system. As it is, the most complete set is characterized by a modulus of elasticity, which was not approximated with a great deal of accuracy (Fig. 5) (60). More importantly, the purpose of the present work was to predict radii, wall thicknesses, and material stiffnesses, and adaptive rules predicting nonlinear geometries and material stiffnesses would require far more than the six parameters in Eq. 8. The present work also reduced pulsatile hemodynamics into a one-dimensional description using the transmission line equations (Eqs. 1–3) to avoid the compounding complexity of solving the more complete Navier-Stokes Equations. This characterization is less accurate in curved segments (as in the aortic arch) and at bifurcations. The present work also reduced vascular stresses into a single endothelial shear stress and a single circumferential wall stress (Eqs. 4–7) to avoid the compounding complexity of modeling stresses within the vessel wall (17). The stress within a thick vessel wall can vary from the intima to the adventitia, depending on the spatially varying material properties of the various layers and residual stress (12). Although residual stress tends to normalize the stress distribution, Laplace's law is exact when used to calculate the average circumferential wall stress in a uniform thick-walled cylinder (9, 12). Nonetheless, the average wall stress is likely to be predicted much poorer in the area of bifurcations and complex geometries such as the aortic arch. The use of the first-order approximations is typically invoked to deal with incomplete empirical data, incomplete understanding of fundamental processes, or lack of computing power. Although the complexity inherent in the present work confronted all three challenges, the primary reason for restricting analysis to first-order approximations was that higher-order descriptions were incompatible with our assumed first-order adaptive rules embodied in Eq. 8.
Complicated assumptions are not necessary to yield complex results.
By limiting the assumptions necessary to create our complex, adaptive model, three emergent behaviors intrinsic to the systemic arterial system were identified: 1) hemodynamics: heterogeneity in regional pressures and flows emerged from local adaptation of arteries to their unique local mechanical environment without the need of global coordinating mechanisms (e.g., hormonal or neural control); 2) biomechanics: heterogeneity in mechanical properties emerged from fundamental rules describing hemodynamics and adaptation without the need to assume a particular a target equilibrium stress a priori (i.e., a set point); and 3) mechanobiology: heterogeneity in arterial structures and mechanical properties emerged without prescribing a heterogeneous adaptive response. All three behaviors are evident in Fig. 4. Assuming each vessel has an identical adaptive response, the equilibrium stresses and mechanical properties vary due to its unique location in the network and its interaction with surrounding vessels. Complexity was not assumed but emerged.
The balance point approach integrates hemodynamics, biomechanics, and mechanobiology.
There has been a strong and growing interest in bridging the typically isolated fields of pulsatile hemodynamics, biomechanics, and mechanobiology to relate altered hemodynamic variables to normal and pathological structures of elastic arteries (8, 17, 57, 58), networks of microvessels (37–41), and networks of conductance vessels (23, 43, 44, 62). The present work is the first to simultaneously predict equilibrium vascular structure, pulsatile pressures and flows, and mechanical stresses in a human systemic arterial system (Fig. 2). Recapitulating the general principle of Cecchini et al. (5), the present work, like that of Liao and Kuo (22), illustrates that a balance point approach (Fig. 3) makes the assumption of a set-point stress entirely unnecessary. A useful result of eliminating the common assumption of a stress set point is that adaptive responses affecting radii, wall thicknesses, and elastic moduli can be related to parameters directly related to mechanobiology (ro, α, ho, β, Eo, and γ). Furthermore, shifts in equilibrium variables may be predicted from alteration in arterial adaptive processes (manifested by changes in sensitivities α, β, and γ). External perturbations (e.g., reduced cardiac output) may affect both mechanical properties and stresses within the network (manifesting as changes in the mechanics curves in Fig. 3). Simultaneous changes in vascular mechanical properties, hemodynamic variables, and stresses due to altered adaptive response or external perturbations are, however, well beyond the scope of the present work.
From “design principles” to “self-assembly principles.”
A useful result of using a balance point approach is that the teleological argument used by West et al. (59) to explain the complex structure of the systemic arterial system can now be related to specific mechanisms leading to complex adaptive behavior. Instead of invoking teleological design principles characterizing the goals of adaptation (28, 59), our mechanistic approach identifies “self-assembly principles” characterizing the means. 1) Stability: independently adapting vessels must achieve stable equilibrium despite their complex interactions. 2) Efficiency of information transfer: a relatively limited number of adaptive rules encoded within genes must generate thousands of arterial structural properties. 3) Adaptability: the adaptive mechanisms that give rise to observed equilibrium properties must also allow the system to adapt to maintain proper function in response to perturbations. This third principle will be the focus of our future work, because this new framework makes it possible to explore a wide array of adaptive responses to perturbations affecting local and global hemodynamics, mechanical stresses, or mechanotransduction, such as increased peripheral resistance, reduced ejection fraction, or aortic coarctation.
GRANTS
This work was supported by
DISCLOSURES
No conflicts of interest, financial or otherwise, are declared by the author(s).
AUTHOR CONTRIBUTIONS
Author contributions: P.H.N., S.F.C.-K., M.W.M., E.T., and C.M.Q. conception and design of research; P.H.N., S.F.C.-K., M.W.M., E.T., and C.M.Q. analyzed data; P.H.N., S.F.C.-K., M.W.M., E.T., and C.M.Q. interpreted results of experiments; P.H.N., S.F.C.-K., M.W.M., and C.M.Q. prepared figures; P.H.N., S.F.C.-K., M.W.M., and C.M.Q. drafted manuscript; P.H.N., S.F.C.-K., M.W.M., E.T., and C.M.Q. edited and revised manuscript; P.H.N., S.F.C.-K., M.W.M., E.T., and C.M.Q. approved final version of manuscript.
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