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Am J Physiol Heart Circ Physiol 293: H3659-H3672, 2007. First published October 19, 2007; doi:10.1152/ajpheart.00712.2007
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Theoretical and electrophysiological evidence for axial loading about aortic baroreceptor nerve terminals in rats

Bin Feng,1,2 Bai-yan Li,1 Eric A. Nauman,2 and John H. Schild1

1Department of Biomedical Engineering, Indiana University Purdue University, Indianapolis; and 2Weldon School of Biomedical Engineering, Purdue University, West Lafayette, Indianapolis

Submitted 19 June 2007 ; accepted in final form 12 October 2007


   ABSTRACT
TOP
 ABSTRACT
MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
Arterial baroreceptors are essential for neurocirculatory control, providing rapid hemodynamic feedback to the central nervous system. The pressure-dependent discharge of carotid and aortic baroreceptor afferents has been extensively studied. A common assumption has been that circumferential deformation of the arterial wall is the predominant mechanical force affecting baroreceptor discharge. However, in vivo the arterial tree is under significant longitudinal tension, leading to the hypothesis that axially directed forces may contribute to baroreceptor function. To test this hypothesis, we utilized a combination of finite element modeling methods and an in vitro rat aortic arch preparation. Model formulation utilized traditional analytic constructs available in the literature followed by refinement of model material parameters through direct comparison of computationally and experimentally generated pressure-diameter curves. The numerical simulations strongly indicated a functional role for axial loading within the region of the baroreceptive nerve terminal. This prediction was confirmed through single-fiber recording of baroreceptor nerve discharge under conditions with and without longitudinal tension in the vessel preparation. The recordings (n = 5) demonstrated that longitudinal tension significantly (P < 0.02) lowered both the pressure threshold (Pth, mmHg) for baroreceptor discharge and sensitivity (Sth, Hz/mmHg). The effect was nearly instantaneous and sustained; i.e., under longitudinal tension average Pth was 84 ± 3 mmHg and Sth was 0.71 ± 0.15 Hz/mmHg, which immediately increased to a Pth of 94 ± 4 mmHg and a Sth of 1.20 ± 0.32 Hz/mmHg with loss of axial tension. Possible explanations of how an abrupt change in axial loading could result in a synchronized increase in afferent drive of the baroreceptor reflex, and the potentiating effect this could have on neurogenically mediated orthostatic intolerance are discussed.

finite element modeling; mechanosensory afferent; orthostatic hypotension; arterial biomechanics; baroreflex

BARORECEPTORS (BRS) ARE MECHANOSENSITIVE nerve endings that reside in the adventitial layer of the aorta and carotid arteries and provide sensory feedback to the central nervous system that is essential for robust control of heart rate and blood pressure (35, 36, 43). Transient increase or sustained elevation of arterial pressure produces mechanical distortion of the BR nerve endings. This can lead to activation of BR sensory pathways, resulting in a rapid reduction in heart rate via the parasympathetic limb of the BR reflex or withdrawal of sympathetic drive. With such a powerful influence over the cardiovascular system, BR function and dysfunction have been implicated in a diverse range of cardiovascular pathologies from neurally mediated syncope and dysrhythmias to acute and chronic hypertension (10, 20, 27). Unfortunately, a clarifying understanding of the BR mechanotransduction complex is lacking. As a result, there remains considerable controversy in the clinical and basic sciences concerning the acute and long-term impact BR function may have on cardiovascular homeostasis in health and disease (30, 45). This has led to a wide range of experimental methodologies designed to quantify the biomechanical and neural aspects of arterial pressure transduction.

The most widely used preparations for studying BR function involve the carotid artery and the proximal aorta because these two arterial constructs contain the majority of BR nerve terminal endings (36). For in situ and in vitro experiments, the rather cylindrical geometry of the carotid arteries lead to simple, axially cannulated vessel preparations (12, 42). Experimental preparations have shown that variation of axial strain within the physiological pressure range is orders of magnitude smaller than that of circumferential strain, which has led to the natural assumption that circumferential deformation plays the dominant role in the mechanotransduction of carotid BRs (14). The ascending aorta and proximal arteries, however, present a far more complex vessel anatomy and hemodynamics than the carotid sinus. For in situ preparations, the arteries are often freed from surrounding tissues and cannulated and for in vitro preparations, a common simplifying approach being to create a linear flow path by cannulating the brachiocephalic artery for inlet and the descending aorta for outlet flow (4, 6, 7). The rational for creating a linear flow path was an attempt to deemphasize any role axial deformation may play in the neural activation of the BR nerve endings. Interpretations of BR function in response to changing intraluminal pressures could therefore center on the circumferential deformation of the arterial wall. However, because such a configuration so markedly deviates from normal vessel orientation and physiology, the possibility exists for experimental bias. Indeed, we believe that the curvilinear anatomy of the aortic arch, the location of branching proximal arteries, and the focal location of aortic BR nerve endings within the junction of these arteries (28) make it imperative that axial deformation of the BR nerve endings be considered.

To explore this possibility, we have developed an anatomically consistent three-dimensional finite element (FE) model of the rat aortic arch and branching arteries. The arterial wall is modeled as hyperelastic materials with appropriate strain energy functions. Empirical derivation of material parameters was carried out using experimental data from our laboratory and the literature (2, 21, 47). The material model captures the nonlinear elastic characteristics of arterial tissues across a large strain domain. Boundary conditions were assigned to the FE model to account for the intrinsic axial tension known to exist within the intact arterial system, especially about the ascending and descending branches of the aortic trunk (21, 47). Numerical results from FE load analyses that mimic our in vitro experiments show marked changes of axial strain in the focal location of the BR terminals. These analytic observations support our hypothesis that throughout the range of physiological pressures axial deformation of the artery may play an important role in defining the dynamic mechanosensing characteristics of aortic BR endings. Interestingly, the lack of intrinsic axial tension of the aorta and branching arteries resulting from in vitro configurations with a straight-line flow path tends to drive the aortic BR receptive region into axial compression, thus offering further support for experimental bias. In contrast, an anatomically correct curvilinear flow path for the aortic arch imparts axial elongation within the BR receptive region. Single-fiber recordings of pressure-dependent discharge confirmed that axial tension has a significant impact on BR function, significantly lowering both pressure threshold (Pth, mmHg) and sensitivity (Sth, Hz/mmHg).


   MATERIALS AND METHODS
TOP
 ABSTRACT
 MATERIALS AND METHODS
RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
Ten adult male Sprague-Dawley rats (Harlan Sprague Dawley, Indianapolis, IN), 49–63 days old and weighing 200–275 g, were randomly divided into two groups of five animals. One group was used to establish a database of key anatomic measurements from the aortic arch and branching arteries, whereas the other group was used for quantification of vessel deformation as a result of computer control of intraluminal pressure. Data resulting from these studies were used in the development of an anatomically consistent FE model of the aortic arch and branching arteries. Subsequent testing and validation of the mathematical formalisms and empirically derived material properties was carried out using computational simulations that accurately reflected the conditions of the in vitro experimental protocols. All animal usage was approved by the Institutional Animal Care and Use Committee of the Indiana University Purdue University School of Science.

Anatomic measurements. Anesthesia was carried out through intraperitoneal injection of a cocktail containing ketamine (87.7 mg/ml) and xylazine (12.3 mg/ml) dosed at 0.1 ml/100 mg body wt. Following loss of consciousness and reflex response, the rat was secured to a platform with the head positioned for surgical exposure and cannulation of the trachea. Artificial ventilation using room air was set to provide respiration rate of 85/min and tidal volume of 1.2–1.5 ml (model 683, Harvard Apparatus). The heart and aortic arch were exposed by midline incision through the sternum. The tissues surrounding the aortic arch, right common carotid, left common carotid, and left subclavian were freed through blunt dissection as was the aortic depressor nerve (ADN) which represents the common nerve bundle that forms from the mechanosensory nerve terminals that arise from the arterial surface. The aortic arch, branching arteries, and ADN were excised intact from the rat and placed into warm Krebs-Henseleit bicarbonate buffer (37°C) containing (in mM): 118 NaCl, 4.7 KCl, 1.25 CaCl2, 1.2 MgSO4, 1.2 KH2PO4, 25 NaHCO3, and 11.1 glucose, bubbled with 5% CO2-95% O2.

Select anatomic features of five aortic arches and branching arteries were measured under stereomicroscopy (x20, Zeiss OMPI-1) using a digital caliper (Marathon). Rings of arterial tissue were trimmed from the arch at select locations for measuring the inner diameters of the arch (note gray bands, Fig. 1). The residual stress within the arterial tissue was assessed by cutting the rings and measuring the opening angles in a manner previously described by Liu and Fung (29). Consistent with the histological studies of Andresen et al. (2) the thickness of adventitia was found to be approximately three-fourths of the media, and so the wall thickness of the FE model was proportioned as 43% for adventitia and 57% for media. Because it has been shown elsewhere that the intima layer makes an insignificant contribution to the bulk mechanical properties of the vessel wall, it was not represented (22, 24).


Figure 1
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  		Fig. 1. Schematic representation of excised rat aortic arch. Top: geometric representation of a rat aortic arch and branching arteries. Letters demarcate specific anatomic regions for measures from each tissue sample. Shaded regions represent regions where rings of tissue were excised to assess opening angle. Table presents averaged measures from 5 rat aortic arches excised and measured in the manner.

 
Construction of geometric model and FE mesh. Using the mean values of the anatomic measurements a three-dimensional computer aided design (CAD) model of the rat aorta, right common carotid (RCC), left common carotid (LCC), and left subclavian (LSC) arteries under load-free conditions was constructed using Pro/Engineering Wildfire (V2.0, Parametric Technology; Fig. 2). To simplify the formation of the FE mesh and increase the computational efficiency of the numerical analyses the aortic arch and branching arteries were assumed to be symmetric across the XY plane. Unless otherwise specified, the XYZ coordinates throughout the manuscript are the same as the Cartesian coordinate shown in Fig. 2.


Figure 2
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  		Fig. 2. Geometric model of the rat aorta under a load-free state. The averaged anatomic measures (Fig. 1) were used to construct a CAD model of the arch and branching arteries. Here geometric symmetry is assumed along the XY plane. The mechanosensitive nerve endings (arrow) are known to be located within the adventitial layer of aortic arch between the left common carotid (LCC) and left subclavian (LSC) arteries with the right common carotid artery (RCC) meant to represent the brachiocephalic region of the right arterial supply before the branch point for the right subclavian artery.

 
The existence of residual stresses within the arterial wall under load-free conditions has been well documented (11). Indeed, when the arch and branching arteries were sectioned along the XY plane, each half tended to expand away from a semicircular geometry toward one that was more consistent with the opening angles of the arterial rings (Fig. 1). Presumably this configuration represented the stress-free state of the arch and branching arteries; i.e., all residual stresses had been completely released. On account of the rather small size and complex anatomy of the rat arch and branching arteries, it was not possible to make consistent sections along the XY plane. Therefore, the stress-free state of the entire arch and branching arteries was represented in the CAD model using the opening angles of the tissue rings from select locations along the arch. The residual stresses were then introduced into the model by deforming the aortic arch from the stress-free configuration to the geometry assumed for the load-free state (Fig. 3, A and B, also see below).


Figure 3
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  		Fig. 3. Construction of finite element (FE) model of the rat aorta. A and B: isometric view of the geometric model under load-free and stress-free states, respectively. C: FE mesh laid down on the geometric profile for the stress-free state of the arch as shown in B. The FE model is divided into 3 distinct anatomic segments: the aortic arch; the LCC, LSC, and RCC branching arteries; and the descending aorta. Each segment is assigned material parameters derived from the literature or empirically determined through testing and measurements (see MATERIALS AND METHODS).

 
The geometric model of the stress-free aortic arch was imported into Hypermesh 7.0 (V7.0, Altair Engineering) and modeled using first-order three-dimensional elements (8-noded brick or 6-noded wedge) for a total of 7,728 nodes and 6,096 elements. Because brick elements pose superior computational accuracy and numerical stability over wedge elements, only 294 wedge elements were utilized in the FE model and only over regions with the least geometric complexity. All other elements were eight-noded bricks. Relatively coarse elements with lateral dimensions in the range of 140–170 µm were assigned to the peripheral regions of the branching arteries and the descending aorta where the vessel anatomy was nearly cylindrical. Considerably finer elements with typical lateral dimension of 50–60 µm were assigned to the regions representing the ascending and curved anatomy of the arch as well as the area about the origins of the branching arteries where the mechanosensory nerve terminals are located (Fig. 3C). Throughout the entire FE model, the thickness of the arterial wall was six elements deep, three for the adventitia and three for the media. Once a final set of constitutive equations and material constants were identified (see below), the selected element size was validated using a simple cylindrical geometry with lateral mesh dimensions of 200, 150, and 50 µm under the same loading conditions used for the in vitro preparation. A comparison of the pressure-diameter relationship revealed a <0.1% difference across all three element sizes.

Mechanical material model, assumptions, constitutive equation and parameter identification. Mechanical tests have shown that arterial wall tissue exhibits somewhat different bulk material properties when comparing measurements made along the longitudinal and circumferential directions (24). Recent computational modeling studies have suggested that this difference may have a limited impact on bulk mechanical properties of biological materials under physiological conditions (38, 39). Therefore, the vessel wall was considered to be a composite of two isotropic hyperelastic materials (adventitia and media), both of which were simulated with a revised Mooney-Rivlin material model (APPENDIX).

The material properties of the arterial tissue were assumed to vary according to position along the aortic arch and branching arteries. Therefore, the FE model was partitioned into three distinct anatomic regions (aortic arch, branching arteries, and descending aorta) and with presumed differences in material properties (Fig. 3C and see below), which was reflected as different sets of material constants (C1, C2, D1, and D2) in the Mooney-Rivlin strain energy density equation. The veracity of treating wall material properties as static is strengthened by experimental evidence demonstrating the lack of any significant change in the pressure encoding characteristics of aortic BRs in the presence of potent vasodilators beyond a modest reduction in the Pth for discharge (1). However, this does not rule out the possibility that myogenic vessel constriction in response to increasing transmural pressures (i.e., Bayliss effect) may have an as yet unrecognized impact on BR function. Considering the experimental observations of Andresen et al. (1) and that our testing protocols involve very slow increases of transmural pressure, we have elected not to represent a myogenic vasoconstriction in the model formulations for the aorta and carotid arteries. The Matlab optimization tool box (V7.2, Mathworks) was used to carry out nonlinear curve fitting to determine two sets of material constants (Table 1) to accommodate both the literature and our experimental data for the carotid and thoracic aortic arteries (Fig. 4A). Details on this parameterization process can be found in the APPENDIX. The nonlinear FE analyses were conducted using a commercial finite element package, ANSYS (V8.1, ANSYS).


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  		Table 1. Material parameters used for the finite element model

 

Figure 4
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  		Fig. 4. Identification and validation of FE model material parameters from the literature. A: FE model performance relating circumferential 2nd Piola- Kirchhoff stress and the stretch ratio across all 3 branching arteries (Carotid, black trace) and the descending aorta (Thoracic, black trace) relative to the corresponding experimental data (Exp) measured from rat carotid and thoracic aorta, respectively (21, 47). B: FE model performance relating the external diameter increase of the aortic arch and slowly ramped (1.3 mmHg/s) intraluminal pressure (Aortic arch, black trace) relative to the corresponding experimental data measured from rat aortic arch (2). The diameter increase was normalized by the initial diameter of the arch at the load-free state, which was 2.38 mm for the experimental data and 2.27 mm for the FE model.

 
FE analysis for identification of aortic arch material parameters. To further refine the material properties of our FE model (Fig. 3), boundary conditions were modified so as to replicate the linear flow path of published data (Fig. 4B) from rat aortic arch (2). Here, the RCC and descending aorta were aligned axially with both ends cannulated; i.e., rigid boundaries and the LCC and LSC ligated as close to the arch as possible. In the FE analysis, intraluminal pressure was increased in increments of 20–200 mmHg, and the diameter of the arch midway between the LCC and LSC was calculated at each pressure level. Empirical adjustment of the relevant parameter sets eventually led to a model-predicted pressure-diameter change that closely approximated the experimental data (Fig. 4B). Confident that an appropriate set of material properties had been identified, the boundary conditions of the FE model were returned to those most representative of the in vitro preparation designed to recreate the physiological load conditions experienced in vivo (i.e., axial stretch, circumferential residual stress, and varying intraluminal pressures).

Functional deformations. The rat aortic arch and branching arteries (n = 5) were harvested following the same procedure for the anatomic measurements. The three branching arteries, i.e., LCC, RCC, and LSC, were ligated at locations 10–15 mm away from their junctions with the aorta. The ascending and descending openings of aortic arch were cannulated using stainless steel tubing with external diameters comparable to the lumen. The arterial tissue was then transferred to a water jacketed chamber and perfused at 2–5 ml/min with a Krebs-Henseleit bicarbonate buffer (37°C) bubbled with 95% O2-5% CO2. To reasonably approximate the natural anatomic configuration of the arterial tissue, the aorta was cannulated as a curved arch and the three branching arteries stretched axially orthogonal to the arch a distance that closely reflected the amount of vessel shortening observed with transection (40–60%).

So as not to evoke the viscoelastic dynamics of the medial smooth muscle intraluminal pressures were slowly ramped (1.3 mmHg/s) using a computer-controlled fluid pump (M6, Intelligent Motion Systems). A proportional-integral-derivative controller was used to regulate pump pressure that was continuously monitored using a silicon strain gauge pressure transducer (Radnoti Glass Technology). Deformation of the arch and three branching arteries was recorded using a digital camera (Guppy F-033B, Allied Vision Technologies) running at 15 frames/s and positioned perpendicularly to the testing chamber. Calibration using the digital calipers made possible the calculation of external vessel diameters using Matlab's image processing toolbox from static camera images. These measurements were later used for further parameterization and validation of the FE model (see RESULTS).

Electrophysiological recording of single-fiber BR discharge. Under stereomicroscopy (x50) the ADN was cleared of any remaining connective tissue and prepared for further manual dissection using 30-gauge needle tips. The common nerve bundle was teased apart lengthwise for several millimeters into two approximately equal-sized segments. One segment was selected for further splitting, which continued to a point where the nerve was subjectively considered to be fine enough for single-fiber recording but thick enough to withstand the mechanical force needed to suspend the fiber across a bipolar Pt-Ir recording electrode (FHC, Bowdoin, ME). The nerve and electrode were raised a few millimeters into a layer of mineral oil that covered the surface of the bathing medium. A slow-pressure ramp was initiated and action potential discharge from the BR nerves was amplified (x20,000, Princeton Applied Research, model 113), digitized (20 kHz, model 118, iWorx/CB Sciences), and displayed in real time on a data acquisition workstation (Dell Optiplex GX270). The nerve splitting and vessel inflation continued until action potential discharge from a few fibers could be visually discriminated. At this point, the stored data were analyzed using a Matlab-based extracellular spike sorting routine that performed a wavelet transformation of each extracellular spike and binned the time series data according to similarity of wavelet coefficients (37). On verification that pressure-dependent discharge from a single BR fiber could be reliably recorded the experimental protocol was initiated and data stored for post hoc analysis. On completion of the testing protocol the time series of pressure-dependent BR fiber discharge was summarized as instantaneous firing frequency (IFF) as a function of ramp pressure. The IFF in Hertz was calculated from the reciprocal of the time between two successive spikes. From this relationship, three distinct measurements were made that captured the essential functional characteristics of each BR (3): 1) the threshold pressure (Pth) for BR nerve discharge calculated as the average pressure over the first 10 spikes, 2) the threshold frequency (Fth) of BR nerve discharge calculated as the average IFF over the 1 s of nerve activity immediately following Pth, and 3) the slope (Sth) of the IFF-pressure relationship calculated over the first 20–30 mmHg immediately following Pth.


   RESULTS
TOP
 ABSTRACT
 MATERIALS AND METHODS
 RESULTS
DISCUSSION
 APPENDIX
 REFERENCES
 
The revised Mooney-Rivlin Strain energy function (Eq. 1) was selected to simulate the nonlinear elastic behavior of rat aortic vessel wall. The rat aorta was divided into three anatomically and functionally distinct regions (i.e., the aortic arch; the branching LCC, LSC, and RCC arteries; and the descending aorta; Fig. 3C). Each region was represented by two different groups of homogeneous materials simulating the adventitial and medial layers, respectively (Table 1). Considerable care was taken to ensure that the material models of the branching arteries and descending aorta provided satisfactory representation of the nonlinear stress-strain characteristics of these vessels as presented in recent literature (Fig. 4).

Additional model verification. Before initiating a series of numerical studies that address our questions concerning the potential mechanical dynamics of the confined anatomic region within which the mechanosensory nerve terminals reside, we sought to further verify the biological realism of the FE model, and material parameters. Using the in vitro arch preparation, intraluminal pressures were slowly ramped (1.3 mmHg/s) from 0 to 180 mmHg. Measures of arch, RCC, LCC, and LSC vessel diameter were made at increments of 10 mmHg from a single image frame and always at the same location. On account of differences in animal age and weight, it was necessary to quantify the data from each recording (n = 5) as a percent increase from the unpressurized vessel diameter. These data from the RCC, LCC, and LSC showed the anticipated sigmoidal increase in diameter albeit with some variability across the five different preparations (Fig. 5). Interestingly, the RCC responded with a greater increase in vessel diameter at lower pressures than the LCC and LSC. But this increase plateaued at ~35% and relatively early in the ramp near 100 mmHg (Fig. 5A). In stark contrast, the increase of external diameter of the LCC and LSC did not plateau until higher intraluminal pressures (120–130 mmHg). The external diameter of the LCC increased to 40–60% as pressure levels raised to 180 mmHg, whereas that for the RCC was somewhat lower at 35–50%. The general pressure-diameter relationship for the LCC and LSC was far more sigmoidal than that for the RCC and much more characteristic of the pressure-diameter relationship of the aortic arch itself (compare Figs. 4C, 5B, and 5C). The FE model was prepared for simulating the same slow ramp of intraluminal pressure. Changes in vessel diameter were quantified directly from the data generated by ANSYS and at a position along the aorta and branching arteries comparable to that made on the in vitro tissue preparation (Fig. 5, note continuous solid traces). These results further confirm that at least under conditions of slow increases in intraluminal pressure our FE model presents a reasonable approximation to the spatial and mechanical features of the corresponding physiological preparation.


Figure 5
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  		Fig. 5. Identification and validation of FE model material parameters: in vitro recordings. FE model performance showing external diameter increase as a percentage of the load-free state for the RCC (A), LCC (B), and LSC (C) branching arteries in response to slowly ramped (1.3 mmHg/s) intraluminal pressure (black trace). For each panel, in vitro measures of external diameter change in response to the same slow ramp of intraluminal pressure (n = 5).

 
Impact of vessel configuration and boundary conditions on bulk deformation. The potential impact of vessel configuration and boundary conditions on tissue deformation was assessed through FE simulations of two distinct in vitro arterial preparations; one was cannulated in such a manner as to create a straight-line flow path that emphasized circumferential deformation (Fig. 6, A, C, and E) and the other a configuration more consistent with the natural anatomic orientation of the arch and branching arteries (Fig. 6, B, D, and F). Contour plots of steady-state von Mises stress were calculated at intraluminal pressures of 40, 100, and 160 mmHg (Fig. 6). As expected, for both configurations the models exhibited substantially greater deformation at 160 mmHg (Fig. 6, E and F) compared with 40-mmHg intraluminal pressures (Fig. 6, A and B). Apart from the flow path the most significant functional difference between the two configurations concerned the axial forces applied to the ends of RCC, LCC, and LSC so that the axial stretch ratios of these three trunks were maintained constant at 1.5 across the range of applied pressures (Fig. 6, B, D, and F). These boundary conditions led to an increased width of the arch along the XY plane, which resulted in increased wall stress about the region of the branching arteries even at 40 mmHg (compare Fig. 6, A and B). This difference became much more apparent at increased pressures where wall stress within the region of the branching arteries was several times greater for the physiological configuration than for the straight-line configuration (note stark contrast between Fig. 6, C and D). Because the material dynamics within this anatomic region are crucial to our understanding of the micromechanical environment potentially experienced by the mechanosensory afferent terminals, further investigation was warranted.


Figure 6
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  		Fig. 6. FE model pressure deformation for both straight-line and physiological flow paths. For the straight-line flow path of Andresen et al. (2). A, C, and E present contours of predicted von Mises stress at intraluminal pressures of 40, 100, and 160 mmHg, respectively. The same sequence of pressures and corresponding von Mises stress for a physiological flow path are presented in B, D, and F. Note the occurrence at lower pressures of greater stress magnitudes (red) at the junctures of the branching arteries and the aorta.

 
Mechanical deformation within the region of the mechanosensory nerve terminal endings. As an initial step toward estimating the potential forces within the immediate region of the mechanosensory nerve terminal endings, the circumferential Cauchy stress and circumferential Hencky strain were calculated under conditions of no load (Fig. 7). Quantifying such initial conditions was particularly important because the FE model of the arch was constructed based on the vessel geometry under stress-free conditions where the arch was allowed to spring open (Fig. 3C). Empirically derived boundary conditions were applied to deform the arch from this stress-free state to a load-free state (Fig. 3A), which made possible the numerical estimation and contour presentation of circumferential residual stress and strain under conditions of no load (Fig. 7A). The Cauchy stress and Hencky strain values were calculated at a point midway between the LSC and LCC, along a straight line through the medial and adventitial layers (Fig. 7A, note dashed line). Note that the medial region extended from the interior of the aorta out to 94 µm at which point the adventitial layer continued to the exterior of the artery at 165 µm from the vessel lumen. Under no-load conditions, the inner layer of the arch was subjected to compression, whereas the outer layer was subjected to elongation (Fig. 7, B and C). As shown in Fig. 7B, the stress magnitude in the adventitial layer was markedly lower than that in the medial layer due to the assumed disparities in material strength.


Figure 7
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  		Fig. 7. FE model predicted circumferential residual stress and strain near baroreceptors. A: a cross section of the circumferential residual stress contour within the region of the mechanosensory terminal endings. These aortic baroreceptors are primarily located within the region between LCC and LSC arteries and have been shown to reside in the adventitial layer, approaching but not penetrating the medial tissues (point "P") (28). B and C: circumferential Cauchy stress and circumferential Hencky strain, respectively, as calculated along the dashed line in A.

 
Clearly, the numerical results under conditions of no load suggested that the mechanosensory nerve terminal endings remain under tension (Fig. 7A), albeit of an insufficient magnitude to elicit neural discharge (see DISCUSSION). A more realistic assessment of physiologically relevant functional loading about the mechanosensory nerve terminal endings was carried out through simulations of slow ramps of intraluminal pressure from 20 to 200 mmHg. Again, these analyses were configured to represent two distinct in vitro arterial preparations; one with boundary conditions for a straight-line flow path (Fig. 6, A, C, and E) and the other with boundary conditions more consistent with the natural anatomic orientation of the arch and branching arteries (Fig. 6, B, D, and F). Estimates of both circumferential and axial Cauchy stress and Hencky strain were calculated as functions of pressure for a specific point within the adventitial layer (Fig. 7A, point "P"). Data for both the straight-line and anatomically consistent boundary conditions were presented for direct comparison (Fig. 8, dashed and solid lines, respectively). Although the circumferential stress for the physiological configuration was consistently 10–12% greater than that for the straight-line configuration, the two pressure-dependent responses closely paralleled one another (Fig. 8A). The pressure-dependent profile for circumferential strain was nearly identical for both configurations (Fig. 8B). A number of differences between the two configurations were apparent along the axial stress and strain profiles (Fig. 8, C and D). At the starting pressure of 20 mmHg, the axial stress for the physiological configuration was approximately four times greater than that for the straight-line configuration. This difference diminished with increased pressures but remained 14% greater at the end of ramp pressure of 200 mmHg. Axial strain showed the most striking differences between the two vessel preparations. At the starting pressure of 20 mmHg, the axial strain for the physiological configuration was more than 10 times greater than that for the straight line configuration. Interestingly, as pressure increased, axial strain decreased for both configurations but with a divergence in magnitude that peaked at nearly a 25 times different around 80 mmHg. As pressure increased from here axial strain for both configurations changed slope and also began to increase (Fig. 8D). With the exception of a slight trend toward convergence at higher pressure, the pressure-axial strain curves from both configurations exhibited parallel profiles. Interestingly, the straight-line configuration essentially remained in axial compression (negative strain) for all but the most extreme pressures, whereas the physiological configuration remained in axial elongation at all pressures.


Figure 8
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  		Fig. 8. Stress and strain within the region of the baroreceptor terminals. FE model calculations of orthogonal stress and strain at point "P" of Fig. 7 as intraluminal pressure was slowly ramped (1.3 mmHg/s) from a steady-state condition of 20–200 mmHg. A and B: circumferential Cauchy stress and circumferential Hencky strain, respectively. C and D: axial Cauchy stress and axial Hencky strain, respectively. For all graphs, dashed lines represent pressure-dependent calculations for a straight-line flow path while the solid lines represent calculations for a physiological flow path.

 
Contour plots of axial Hencky strain were generated from FE analyses simulating the straight-line and physiological configurations at 80 mmHg and within the receptive region of the mechanosensory nerve terminals (Fig. 9, A and B). A comparison of the profiles of the strain contours suggests that the straight-line configuration mostly under axial compression (negative strain), whereas the physiological configuration is always under axial elongation (positive strain).


Figure 9
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  		Fig. 9. Contour of Hencky strain within the adventitial layer. Under a static intraluminal pressure of 100 mmHg, the straight line flow path (A) clearly shows a pattern of tissue compression (negative strain) while the physiological flow path (B) is clearly in tension.

 
Impact of longitudinal tension along the LCC, LSC, and RCC arteries on BR function. A critically important aspect of any modeling study is experimental testing of the analytically based hypotheses arising from the numerical simulations. Here, we investigated the suggestion that longitudinal tension along the proximal arteries imparts a range of pressure-dependent axial deformation about the region in which the aortic mechanosensory terminals reside (28) that is of sufficient magnitude to effect BR function (Figs. 6–9). Extracellular recordings of nerve discharge from single, presumably myelinated BR afferent fibers (n = 5) were carried out using the same pressure ramp protocol used for validating FE model material parameters (Fig. 5 and Fig. 6, B, D, and F). From a conditioning pressure of 60-mmHg longitudinal tension (stretching) was maintained for at least 60 min before initiating the slow pressure ramp protocol. Following completion of the ramp protocol and return to the same conditioning pressure the longitudinal tension within the LCC, LSC, and RCC arteries was released, and the slow pressure ramp protocol was immediately repeated. The unstretched vessel was returned to the conditioning pressure for a minimum of 30 min, and the pressure ramp protocol was repeated for a third time.

The pressure-dependent IFF curves for these two configurations, which were predicted to impart markedly different axial deformation dynamics on the mechanosensory terminal endings (Figs. 6–9), showed striking differences across several characteristic features of BR function (Fig. 10). Most notable were the differences in Pth and Sth, which were both markedly lower under stretched compared with unstretched conditions. For the unstretched condition, this particular BR exhibited an IFF that saturated by ~25 mmHg from Pth, whereas the stretched condition continued to show a positive slope up to the maximum testing pressure of 180 mmHg, albeit at a rate that was somewhat lower than the Sth calculated for this fiber (Fig. 10, legend). Perhaps the most striking observation was that this effect on Pth and Sth was immediate (Fig. 10B) and sustained for >30 min (Fig. 10C); i.e., Sth was unchanged after 30 min, whereas the unstretched Pth decreased by <5 mmHg but remained considerably greater than the Pth for the stretched condition. The combined reduction in these two functional properties resulted in a much broader dynamic range for BR encoding of arterial pressure, which was clearly represented in the pooled (n = 5) data where the stretched condition exhibited averaged Pth (84 ± 3 mmHg) and Sth (0.71 ± 0.15 Hz/mmHg) that were both significantly (P < 0.02) lower than the averaged Pth (94 ± 4 mmHg) and Sth (1.20 ± 0.32 Hz/mmHg) for the unstretched condition after more than 30 min of recovery. Interestingly, both the stretched and unstretched configurations exhibited essentially the same Fth and peak IFF at maximum inflation pressures (R2 > 0.9).


Figure 10
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  		Fig. 10. Impact of axial pretensioning on frequency-dependent discharge of aortic baroreceptors. Comparison of single baroreceptor fiber discharge with (A) and without (B and C) longitudinal pretensioning of the LCC, LSC, and RCC branching arteries. A: imparting a physiologically relevant length of longitudinal stretching of these arteries markedly reduced the threshold pressure for initial nerve discharge and baroreceptor sensitivity (Pth of 88 mmHg and Sth of 0.63 Hz/mmHg) compared with the IFF-pressure characteristics of the same baroreceptor fiber but in an unstretched configuration. B: immediately after release of longitudinal pretensioning (Pth of 102 mmHg and Sth of 1.65 Hz/mmHg). C: after a recovery period of >30 min at the conditioning pressure of 60 mmHg (Pth of 97 mmHg and Sth of 1.63 Hz/mmHg).

 

   DISCUSSION
TOP
 ABSTRACT
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
APPENDIX
 REFERENCES
 
The central objective of this study was to test the hypothesis that tissue deformation about the region of BR innervation of the rat aortic arch is significantly influenced by the manner in which the vessel is prepared for in vitro or in situ study. In particular, configurations that fail to adhere to the normal anatomic flow path and orientation of the arch and that result in release of the inherent longitudinal tension within the arch and branching arteries (25) may unduly bias the BR terminal endings toward mechanosensing of circumferential deformations while reducing the potential impact of axial loading. To quantitatively investigate this possibility, we utilized a combination of traditional FE modeling methods and an in vitro rat aortic arch preparation, which both maintain physiological consistency with the vessel orientation in vivo. Namely, an anatomically consistent orientation of the arch and flow path, recognition of residual stresses within the arterial wall, and longitudinal tensioning of the LCC, LSC, and RCC arteries (Figs. 1–3). Identification of FE model material parameters was carried out using analytic methods and data from the literature followed by parameter refinement through direct comparison of numerical and companion intraluminal pressure-vessel diameter curves (Figs. 4 and 5). Subsequent numerical studies were carried out to investigate vessel deformation for both an unloaded, straight-line flow path and an anatomically consistent flow path with the arch and branching arteries under realistic levels of longitudinal tension. These analytic results supported our hypothesis that axial loading within the region of the mechanosensitive terminal endings can have a marked impact on the pressure-dependent deformation of the arterial wall (Figs. 6–9). Our companion electrophysiological studies demonstrated that such axial loading dramatically expands the functional range of BR neural encoding of arterial blood pressure (Fig. 10).

Model formulation, validating assumptions, and limitations. The longer term objective of our modeling studies is to investigate the local micromechanical environment about the mechanosensory terminals of BR afferents. This study enables us to estimate the regional boundary conditions about these nerve endings from an investigation of bulk tissue deformation. Details of model development and assumptions are guided by the physical characteristics of, and constraints associated with, the in vitro rat aortic arch preparation used for both refinement of model material parameters and testing of model-generated hypotheses concerning BR function under a very specific set of experimental conditions. A higher order material model would be required to consider the complex dynamics associated with ramp pressures much faster than the 1.3 mmHg/s utilized in our experiments or pulsatile flow. The analytic formulations presented here can provide the boundary conditions for a FE model with sufficient spatial resolution to accommodate the subtle neuroanatomic features of the BR terminal endings revealed by three-dimensional confocal images from a collaborator's laboratory (18). Realization of an ion channel-based model capable of addressing issues of mechanosensory excitability will benefit from simulations of the micromechanical forces about the afferent terminal ending.

Two simplifying assumptions are made during the design of the FE model of the arch. First, the arterial vessel wall is assumed to be an ideal elastic material. Viscous characteristics are not considered because all computational studies and in vitro testing of the numerical results are carried out under quasi-static loading conditions (i.e., 1.3 mmHg/s). For such a slow increase in intraluminal pressure, the viscous properties of arterial tissue are assumed to have a negligible impact on material dynamics. This has been demonstrated in previous studies of the rat aortic arch where a similar slow ramp pressure protocol evoked nearly identical pressure-diameter curves across consecutive loading cycles (33). Such evidence is supportive of our assumption that an elastic material model is adequate to capture the nonlinear mechanical properties of the aorta and branching arteries under these specific in vitro experimental conditions. Assuming that measures of the biaxial stress-strain relation of aortic arch tissue are available, a higher order analytic representation would be more biologically realistic but untestable given the practical constraints of our experimental preparation. Similarly, these assumptions preclude the ability to represent any changes in material properties potentially arising from an intrinsic myogenic response from the aorta and carotid arteries.

Second, as has been successfully implemented in related studies, both the adventitial and medial layers of the arterial wall are assumed to be isotropic (38, 39). The implications of such an assumption must be carefully considered as the material properties about the aortic arch may very well be inhomogeneous with unique values in the areas of branching. The benefits of anisotropic constitutive models of arterial wall tissues have been demonstrated (16, 49). However, such analytic complexity often precludes the use of commercially available FE modeling platforms, and the solutions often exploit anatomic symmetries that permit simplification of vessel geometry. These simplifications would not be possible given the curvilinear structure of the arch and branching arteries. Still, an isotropic representation imparts an unknown level of numerical inaccuracy concerning the circumferential and axial stress and strain about the region of the mechanosensory terminal endings. It is quite possible that the material properties could be inhomogeneous and that there may be unique values in the areas of branching. Therefore, model results must be carefully interpreted relative to the associated experimental platform and loading conditions. Given this caveat and that arterial wall tissue is far more compliant in the axial compared with the circumferential direction (24) an assumption of isotropy likely underestimates the extent of axial deformation, particularly given the quasi-static loading conditions.

Given that our biomechanically related questions concerning BR function are confined to a relatively restricted region of aortic tissue, it is essential that the bulk model adequately represent residual wall stress, an estimate for which was made by measuring the opening angle of narrow rings of tissue cut from along the length of the aorta (Fig. 1). Once the bands are cut and allowed to expand the residual stress can be assumed to be completely released (17). Our measures of opening angle are consistent with similar measures from other investigations of aortic tissues (29, 40). As reported in these earlier studies, there is an overall increase in opening angle along the descending portion of the arch (Fig. 1, bands C, D, and E), but much larger opening angles are noted over the ascending region approaching the proximally branching arteries (Fig. 1, bands A and B). Collectively these data imply that the circumferential residual stress and strain within the region where BRs are located may be somewhat smaller than other regions along the aortic arch but still of sufficient magnitude to potentially impart local tension about the mechanosensory terminal endings at very low intraluminal pressures (Fig. 8, note offsets at low pressures). These opening angles help define the construction of the arch CAD model by assuming the opening angles at the stress-free state would be identical to that measured from the narrow bands of aortic tissues (Fig. 1). The circumferential residual stress is then incorporated into the model by deforming the vessel from a stress-free (Fig. 3C) to load-free state that matched the averaged physical dimensions of the excised aortic tissue samples (Fig. 1).

The nonlinear elastic properties of both the medial and adventitial layers are synthesized according to the hyperelastic material law using a revised Mooney-Rivlin strain energy function (Eqs. 2 and 3). These formalisms ensure that the classic nonlinear sigmoidal-shaped stress-strain relation for arterial vessels formed the basis for refinement of material parameters using a combination of published and primary experimental data (Fig. 4). These efforts led to a final set of material parameters capable of capturing the unique pressure-diameter profiles of the aortic arch and branching arteries (Table 1, Fig. 5). Evidence of the markedly different mechanical material properties between the carotid artery and thoracic aorta (Fig. 4A) strongly suggests a gradient of material inhomogeneity along the curvilinear anatomy of the aortic arch. Representing the vessel complex as two layers spread across three regions (aortic arch, branching arteries, and descending aorta), all with distinct material properties, is a necessary compromise (Fig. 3C). Although this inevitably results in junction effects in the FE analysis (e.g., note discontinuities in Figs. 6 and 7, B and C) the loading paradigms are of sufficiently low rate so as to ensure convergence and accuracy throughout the numerical simulations.

Further confidence in the model formulation comes from a direct comparison of numerical result with measures of pressure-dependent changes in the diameters of the branching arteries (Fig. 5). The pressure-diameter curve for the RCC is quite different from those for the LCC and LSC, exhibiting a near saturating response at pressures not much greater than 100 mmHg. This is consistent with the fact that the RCC was shown to have a larger diameter-to-thickness ratio (Fig. 1), which, in accordance with the law of Laplace, resulted in larger average circumferential stress across the vessel wall under same intraluminal pressure. Therefore, in response to the same ramped pressure, the circumferential strain in RCC should be expected to exhibit a stress-strain relationship that approaches a saturated response at lower pressure than for the LCC and LSC. This is well reflected in both the experimental and numerical results where the diameter increase for the RCC plateaus at pressures (90–100 mmHg) somewhat lower than that of LCC and LSC (120–130 mmHg). Also as a result of this delayed plateau, the LCC and LSC exhibits a greater circumferential deformation at higher pressures (>130 mmHg) and along an asymptotic trajectory with greater slope than observed for the RCC.

Impact of vessel orientation and axial loading of proximal arteries. In situ BR preparations that involve direct control over flow necessitate vessel cannulation. This relaxes axial tension and frees the artery from surrounding tissues, "unloading" the vessel walls from a supporting superstructure (6, 7, 41). In vitro preparations most often cannulate the excised arch in a straight-line manner so as to emphasize the circumferential deformation of the arterial wall (2, 4, 6, 7, 12, 32, 42). The similar loading characteristics and vessel preparation has yielded single fiber data that are quantitatively similar across preparations and species. A comparison of contours of the von Mises Stress, a scalar measure of the stress state (the normal and shear stresses) at any point within the vessel, as a function of intraluminal pressure suggests that a straight-line flow path or a release from longitudinal tension results in far lower stress magnitudes than for the physiological flow path (Fig. 6). A cross-sectional view of the interior of the aorta within the region of the BR nerve terminals suggests positive (tensile) loading even without an inflation pressure on account of the residual stresses within the arterial wall (Fig. 7). Because the medial layer is 10 times stiffer than the adventitial layer (Table 1), it is reasonable that there is about an order of magnitude higher stress level in the contour than for the circumferential Cauchy stress (compare stress magnitudes of Figs. 6 and 8).

The simulation results for axial strain for both the straight-line and physiological configurations exhibited rather unique biphasic profiles (Fig. 8D). From the start of the slow pressure ramp axial strain decreased until intraluminal pressures reached ~80 mmHg, at which point axial strain increased with increasing intraluminal pressures. This phenomena was observed for both the straight-line and physiological configurations but the former was essentially in axial compression (negative strain) throughout most of the pressure range. The biphasic characteristic occurred as a result of two competing factors during the ramped pressure loading: axial extension of aorta and axial compression arising from the LCC and LSC branching arteries. Over the lower range of pressures (0–80 mmHg), the LCC and LSC showed marked dilation (Fig. 5), which resulted in axial compression within the region between both arteries. In the meanwhile, the axial elongation caused by the inflation of aorta was shadowed by the compressive effects until ~80–100 mmHg. The circumferential dilation of the LCC and LSC started to lessen as pressures continued to increase beyond 100 mmHg where the axial extension of the aorta began to dominate, resulting in a gradual shift toward increasing axial strain.

Functional impact of axial pretensioning upon baroreception. A direct comparison of the stress and strain magnitudes may lead one to conclude that circumferential deformation was the dominant factor underlying the mechanotransduction of BR afferent terminal endings. However, it appeared that longitudinal tension associated with the physiological configuration effectively biased the local tissue deformation toward tensile as opposed to compressive forces (Fig. 8D). This model prediction is consistent with the characteristics of single-fiber discharge with and without longitudinal tension along the proximal branching arteries (compare Fig. 10, A and B). Axial pretensioning dramatically increased the pressure encoding range of the BR afferents as evidenced by the significantly lower slope of the IFF-pressure curve, i.e., an average Sth of 0.71 ± 0.15 Hz/mmHg with longitudinally stretching compared with a Sth of 1.20 ± 0.32 Hz/mmHg on release of axial tension. However, by far the most striking experimental observation was the rapidity with which the loss of longitudinal pretensioning resulted in a significant change in Pth and Sth. An effect that remained essentially unchanged after at least 30 min at a fixed inflation pressure of 60 mmHg (Fig. 10, B and C). Such a response was in stark contrast to the well recognized phenomena of acute resetting which can achieve a stable operating range within several minutes of exposure to a new mean arterial pressure (i.e., circumferential deformation), with Pth shifting in the direction of the conditioning pressure but with Sth remaining unchanged (32). These observations may be indicative of a rigid, nonelastic tethering of the BR terminal along the longitudinal axis of the vessel, one that does not exhibit conventional viscoelastic properties. This would be consistent with the experimental evidence that resetting of the BR reflex may not be complete and that arterial BR actively participate in the long term control of blood pressure (45). Characterization of the impact prestretched conditions may have on the relationship between mean conditioning pressure and BR resetting will require further single-fiber recordings. However, a reduction in Sth and the concomitant broadening of the pressure encoding range could reasonably be expected to impact the ratio of BR resetting (4, 5).

Potential implications of a rapid reduction in BR Pth and Sth. Although the present study strongly suggests a prominent role for axial loading in defining the operational dynamics of aortic BRs, only extreme conditions have been investigated, i.e., stretched or unstretched. The functional significance of this observation is bolstered by recent in situ evidence from a porcine model that the length of a segment of artery changes during the cardiac cycle (46). Our model lacks the analytic complexity to quantify axial strain dynamics over such rapid rates of pressure variation. But our experimental observations of a rapid and sustained change in Pth and Sth with changing longitudinal tension coupled with such evidence of beat-to-beat changes in vessel length raise the possibility that these functional properties of BR may vary over the course of the cardiac cycle. This would add an as yet unappreciated dimension to the neural encoding properties of these pressure-sensitive afferents. However, what is clearly demonstrated in both the theoretical and experimental results is that changes in longitudinal tension rapidly and robustly reduce BR sensitivity for encoding mean or slowly varying (<1.3 mmHg/s) arterial pressures. A reduction in afferent sensitivity has long been considered an important potential mechanism associated with measured reductions in cardiovagal BR sensitivity. Our single-fiber recordings therefore provide indirect evidence of a potential role for axial loading as a biomechanical substrate for neurogenically mediated orthostatic intolerance. Autonomic studies have shown that spontaneous cardiovagal BR sensitivity is significantly reduced over the course of transition from a supine to an upright posture (34). Furthermore, noninvasive measures of cardiovagal tone and BR sensitivity significantly decreased with head-up tilt and lower body negative pressures, and these indexes strongly correlated with symptoms of presyncope in a group of orthostatic intolerant subjects (26). Transition from a supine position to an upright posture could be expected to increase longitudinal tension along the arterial tree. Such an effect on hemodynamics would be similar, in principle, to theories that orthostatic intolerance may be due to increased loading of ventricular mechanoreceptors (19). In addition to a reduction in sensitivity, our data strongly suggest that transition to an upright posture may impart an instantaneous shift toward a lower Pth (Fig. 10B). Should this occur for a significant percentage of aortic BRs, the net effect would be a synchronized increase in the afferent drive of the arterial BR. In situ studies of the BR utilizing electrical stimulation of the aortic depressor nerve have clearly demonstrated that such a synchronized step increase in BR fiber discharge causes dramatic reductions in heart rate and mean arterial pressure within a few cardiac cycles (15). Similarly, in human subjects, postural hypotension and bradycardia can arise from such abnormal activation of the BR reflex (31).

Such an interpretation, although requiring further experimental clarification, is consistent with the strong clinical evidence that neurally mediated orthostatic intolerance can be effectively managed through physical maneuvers (e.g., leg crossing, muscle tensing, and squatting) that increase intrathoracic pressures and thereby limit the severity of the vasovagal event (9).

Conclusion. Our working hypothesis for this integrated numerical and in vitro study has been that the curvilinear anatomy of the aortic arch, the location of branching proximal arteries, and the restricted anatomical distribution of mechanosensory nerve terminal endings within the junction of these arteries collectively bring about sufficient conditions for axial deformation to play a role in BR function (Figs. 1–3). Development of FE model formalisms proceeded in a manner that maintained good agreement with published data and our own experimental measures of pressure-dependent changes in vessel diameter (Figs. 4 and 5), strengthening the physiological significance of model predicted pressure-dependent force distributions across the aortic arch and within the region of the BR terminals (Figs. 6, 7, and 9). Although the relative magnitudes of axial stress and strain over the physiological range of arterial pressures are predicted to be substantially less than those in the circumferential direction (Fig. 8), the functional impact of axial stress and strain on the pressure-dependent discharge of BR afferents (Pth and Sth) is both robust and dramatic (Fig. 10).


   APPENDIX
TOP
 ABSTRACT
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
REFERENCES
 
The medial and adventitial layers were both considered to be isotropic and incompressible hyperelastic materials. A revised Mooney-Rivlin model was selected to describe the nonlinear elastic behavior of rat arterial tissue with a strain energy density function of the form (44):

Formula
Note that the first, second, and fourth terms of Eq. 1 formed the conventional two-parameter Mooney-Rivlin model. However, to achieve an adequate fit to the sigmoidal-shaped stress-strain relation of arterial tissues, an additional exponential term was added to the strain energy function (23). In the fourth term of Eq. 1, J is the ratio of the deformed elastic volume to the reference (undeformed) volume and defined as: J = Formula, where I3 is the third strain invariant. The first (I1) and second (I2) deviatoric strain invariants expand as:

Formula

Formula
where I1 and I2 represent the first and second strain invariants, respectively. The bulk modulus (K) imparted material incompressibility and is defined as:

Formula
where {nu} represents Poisson's ratio assigned here a value of 0.49. Thus the bulk modulus K was at least two orders of magnitude greater than parameters C1 and C2, which ensured that the material could be approximated as nearly incompressible (8).

Material Parameter Identification for Branching Arteries and Descending Aorta

The relation between the circumferential 2nd Piola-Kirchhoff stress (PK2 stress) and the circumferential stretch ratio has been studied extensively through material tests on rat arterial tissues (21, 47). Following the assumed strain energy density function in Eq. 1, this relationship for rat carotid and thoracic aorta was also calculated mathematically under the assumption that the blood vessel walls were homogeneous across the thickness (Fig. 4A). This permitted application of the fully incompatibility condition, and as a result the ratio of volume change J became unity and Eq. 1 simplified as:

Formula
This strain energy function was used to derive the circumferential PK2 stresses as a function of circumferential stretch ratios:

Formula
S is the PK2 stress and {lambda} is the principle stretch ratio with subscripts 1 and 2 denoting circumferential and axial directions, respectively, for all terms.

The Matlab optimization tool box (V7.2, Mathworks) was used to carry out a nonlinear curve fitting to determine two sets of material constants for the terms C1, C2, D1, and D2 of Eq. 3 to provide best fit to the literature data for the carotid and thoracic aortic arteries. Consistent with the lamina theories of composite materials, these two parameter sets described the equivalent mechanical properties of a composite material that consists of two components: adventitia and media (13). Because the elastic modulus of media is approximately one order of magnitude higher than modulus of adventitia, the strain energy density arising from the media layer was assumed to be 10 times of adventitia layer (22, 48), which provided the following composite relationships for the medial and adventitial layers:

Formula

Formula

Formula

Formula
Here, the volume content of the composite for the media (p) and adventitia (q) were 57 and 43%, respectively.(2) The material model of the carotid was assigned to all three branching arteries with the assumption that the mechanical properties of subclavian were similar to those of the carotid (Table 1).


   ACKNOWLEDGMENTS
 
This work was supported by National Heart, Lung, and Blood Institute Grant HL-072012 and Research Support Funds Grant support from Indiana University Purdue University Indianapolis to J. H. Schild. Partial support for B. Feng derived from National Science Foundation Integrative Graduate Education and Research Training Program Grant in Therapeutic and Diagnostic Devices DGE-99-72770.


   FOOTNOTES
 

Address for reprint requests and other correspondence: J. H. Schild, Dept. of Biomedical Engineering, Indiana Univ. Purdue Univ. Indianapolis, 723 W. Michigan St., SL174, Indianapolis, IN 46202 (e-mail: jschild{at}iupui.edu)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.


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 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
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