The biaxial active mechanical properties of the porcine primary renal artery

Abstract

The mechanical response of arteries under physiological loads can be delineated into passive and active components. The passive response is governed by the load-bearing constituents within the arterial wall, elastin, collagen, and water, while the active response is a result of vascular smooth muscle cell (SMC) contraction. In muscular blood vessels, such as the primary renal artery, high SMC wall content suggests an elevated importance of the active response in determining overall vessel behavior. This study is a continuation of our previous investigation, in which a four-fiber constitutive model of the passive response of the primary porcine renal artery was identified. Here we focus on the active response of this vessel, specifically in the case of maximal SMC contraction, and develop a constitutive model of the active stress–stretch relations. The results of this study demonstrate the existence of biaxial active stress in the vessel wall, and suggest the active mechanical response is a critical component of renal arterial performance.

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).