Research PaperThe biaxial active mechanical properties of the porcine primary renal artery
Introduction
Arteries under physiological loads exhibit a complex mechanical response that is governed by the geometrical dimensions of the vessel and mechanical properties of arterial tissue. The major load-bearing constituents of the arterial wall, elastin, collagen and water, determine the so called passive mechanical properties of vascular tissue. These constituents ensure integrity of the arterial wall and performance of arterial physiological function as a blood conduit and elastic buffer that reduces the cardiac preload. The smooth muscle cells (SMCs) in the arterial wall have an insignificant effect on the passive properties, but when appropriately activated by mechanical, electrical, or chemical stimuli, they can contract or relax and in turn constrict or dilate the vessel. This phenomenon is termed the active mechanical response. Under normal physiological conditions the SMCs are partially contracted and the vessel manifests basal muscular tone. In concert with the residual strains, muscular tone synergistically contributes to homogenization of the strain and stress distribution across the arterial wall and thus promotes a preferable local mechanical environment for vascular cells (Rachev and Hayashi, 1999, Matsumoto et al., 1996). Moreover, the active response occurs as an acute primary mechanism directed to cope with short term changes in flow rate and/or arterial pressure (Brownlee and Langille, 1991, Bayliss, 1902).
The passive and active mechanical response of arteries can be described in several manners. At the overall arterial level, the response is illustrated by pressure–diameter and axial force-axial stretch relationships via data points in the corresponding 2-D planes (Cox, 1975, Cox, 1978, Dobrin, 1978). Often as complementary descriptors, several linearized measures in the vicinity of a particular deformed state are calculated based entirely on experimental data, such as compliance and pressure–diameter modulus (called also Peterson’s modulus) (Peterson et al., 1960). After appropriate data processing, the mechanical response can be described via stress–strain (or stretch) relationships, which characterize the properties of the arterial tissue. Stresses and strains are calculated after adoption of certain assumptions for the deformation process.
One of the basic tasks of vascular biomechanics is the quantification of the mechanical properties of arteries in terms of continuum mechanics-based constitutive equations. Completion of this task enables the formulation and solution of boundary value problems that provide predictive results for the mechanical response of arteries and for calculation of the stress and strain distribution across the arterial wall. The stress and strain fields determine the local mechanical environment of the vascular cells, the mechanobiological response of which governs arterial homeostasis. Constitutive equations also provide a theoretical framework for design of experimental investigations and processing data from mechanical tests. While the passive mechanical properties of arteries have been thoroughly investigated (Vito and Dixon, 2003, Holzapfel and Ogden, 2010), there are less published studies on mathematical modeling of the active response.
To our knowledge the first continuum-based constitutive model that accounts for the effects of SMCs was proposed in (Rachev and Hayashi, 1999). The model is based on the assumption that when appropriately stimulated, the SMCs produce an active circumferential stress that is additive to the passive stress borne by the extracellular structural constituents. A justification for the stress orientation is the experimental observation that SMCs are aligned mainly in the circumferential direction (Cox, 1978). The magnitude of the active stress is considered dependent on the intensity of stimulation and on the deformed configuration of the artery (Dobrin, 1973, Dobrin, 1983). The model was adopted in subsequent studies that examine the active response of different arterial types, such as basilar artery (Cardamone et al., 2009, Karšaj and Humphrey, 2012, Valentin et al., 2009, Valentin and Humphrey, 2009), and iliac artery (Humphrey and Wilson, 2003). Based on the observation that some SMCs might be oriented not solely in the circumferential direction (Kockx et al., 1993, Dartsch and Hämmerle, 1986), several studies generalized the model by adding an active axial stress (Wagner and Humphrey, 2011, Chen et al., 2013, Agianniotis et al., 2012). The state of knowledge on the active arterial response was summarized recently in a comprehensive review on vascular tissue mechanical models (Kim and Wagenseil, 2014).
Surprisingly there is a lack of constitutive modeling of the active arterial properties of renal arteries. They belong to the class of muscular arteries in which a large portion of the media is occupied by vascular SMCs and the active response is a typical manifestation of arterial performance. An impaired mechanical response, including the active component, is associated with vascular diseases, such as stenosis, hypertension, aneurysm and occlusion, and has increasingly gained recognition as a potential risk factor for kidney failure or cardiovascular morbidity and mortality.
Constitutive equations are quantified from data acquired in appropriate mechanical tests. Briefly, the commonly accepted ex-vivo testing methodology includes the inflation of a tubular specimen performed quasi-statically at fixed levels of axial stretch. First the inflation-extension experiment is run while the SMCs are kept alive and are stimulated to contract. Afterwards, the contractile capability of the muscle is abolished and the mechanical test is repeated. The generated data are processed in the reverse order. First, considering the arterial tissue as an elastic incompressible solid, the passive mechanical properties are determined in terms of a strain energy density function. Next, an analytical form and associated material constants are identified to model the active arterial response. A different approach was recently proposed for constitutive formulation of the axial active stress in mouse aorta (Agianniotis et al., 2012). It allows quantification of the active stress independently of the constitutive modeling of the passive mechanical properties.
This study is a continuation of our previous investigation on the passive mechanical properties of the porcine primary renal artery, which was analyzed in the framework of a four-fiber constitutive model (Zhou et al., 2014). We focus on the description of the active response in the case of maximally contracted SMCs and on the phenomenological constitutive formulation of the active properties by adopting, with some modification, the approach proposed in (Agianniotis et al., 2012).
Section snippets
Mechanical testing
All tissue handling protocols were approved by the Institutional Animal Care and Use Committee at the University of South Carolina. The right primary renal arteries were dissected from freshly harvested adult (7–12 month old) porcine kidneys obtained at a local slaughterhouse (Caughman’s Meat Plant Inc., Lexington, SC). Prior to arterial excision from the intact aorta and kidney, two dots of tissue marking dye were applied close to the proximal and distal ends of the vessel and the distance
Results
Morphometric measurements of the porcine primary renal arteries in the stress-free configuration were obtained via image analysis (Table 1). Geometric variability among arteries might be due in part to the different age (7–12 month old) and weight of the pigs, although animal-specific information was not available for correlation. Experimental results show that complete SMC relaxation causes an increase in the traction-free sample length as compared to when the basal tone is preserved, i.e.
Discussion
This study focuses on the quantification of the active mechanical response of the porcine primary renal artery and constitutive formulation of the mechanical properties of the arterial tissue from in-vitro biaxial tests on tubular specimens. We recently quantified the strain energy function of the porcine primary renal artery using a 3-D approach (Zhou et al., 2014); in this study we again adopted a four-fiber family model but introduce a new 2-D approach. The 3-D approach accounts for a
Conclusion
Our obtained experimental results provide novel information on the biaxial active mechanical response of the primary porcine renal artery. The proposed constitutive model, developed in the framework of cardiovascular solid mechanics, provides theoretical active stress–stretch relations that agree well with experimental data. Our findings extend the current level of knowledge about arterial mechanics in a critical region of the circulatory system and demonstrate a robust experimental/theoretical
Acknowledgements
This work was supported by the National Science Foundation/EPSCoR Grant (EPS-903795) and the National Institute of Health SC/INBRE Grant (P20 GM103499).
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