Comparative study of variations in mechanical stress and strain of human blood vessels: mechanical reference for vascular cell mechano-biology

Abstract

The diseases of human blood vessels are closely associated with local mechanical variations. A better understanding of the quantitative correlation in mechanical environment between the current mechano-biological studies and vascular physiological or pathological conditions in vivo is crucial for evaluating numerous existing results and exploring new factors for disease discovery. In this study, six representative human blood vessels with known experimental measurements were selected, and their stress and strain variations in vessel walls under different blood pressures were analyzed based on nonlinear elastic theory. The results suggest that conventional mechano-biological experiments seeking the different biological expressions of cells at high/low mechanical loadings are ambiguous as references for studying vascular diseases, because distinct “site-specific” characteristics appear in different vessels. The present results demonstrate that the inner surface of the vessel wall does not always suffer the most severe stretch under high blood pressures comparing to the outer surface. Higher tension on the outer surface of aortas supports the hypothesis of the outside-in inflammation dominated by aortic adventitial fibroblasts. These results indicate that cellular studies at different mechanical niches should be “disease-specific” as well. The present results demonstrate considerable stress gradients across the wall thickness, which indicate micro-scale mechanical variations existing around the vascular cells, and imply that the physiological or pathological changes are not static processes confined within isolated regions, but are coupled with dynamic cell behaviors such as migration. The results suggest that the stress gradients, as well as the mechanical stresses and strains, are key factors constituting the mechanical niches, which may shed new light on “factor-specific” experiments of vascular cell mechano-biology.

1 Introduction

It is getting to know that mechanical environments such as stresses and strains play crucial roles in the comprehensive remodeling behaviors of human blood vessels, which are important in both regular physiological functions and pathological dysfunctions. For a better understanding of the mechanism that couples biological and mechanical physics from cellular to tissue scales, numerous in vitro mechano-biological experiments for vascular cells have been reported (Wells 2008; Humphrey et al. 2014; Sears and Kaunas 2016). However, from the viewpoint of the mechanical niches of resident cells (Ingber 2003; Chen 2008), to what extent current studies in vitro on cellular mechano-transduction can correlate quantitatively to vascular physiological or pathological conditions in vivo is a critical question to ask for evaluating existing results and exploring new factors for disease discovery.

Cells as machines respond to their mechanical stimuli appropriately (Sheetz and Yu 2018). According to the classical stress-growth theory (Fung 1991), none of the mechano-biological responses of cells is proportional to even a single mechanical stimulus. Blood vessels exhibit different geometries, material properties, loadings, and residual stresses in vivo at different arterial locations (Saini et al. 1995; Holzapfel et al. 2000; Kamenskiy et al. 2014), which makes the mechanical environments of vascular cells complicated (Azar et al. 2018). Although several mechanical factors, such as the stress, strain, and stiffness of substrates, have been extensively reported as crucial parameters in mechano-biological studies, a fundamental standard for the guidance of mechanical stimuli is still lacking (Kumar et al. 2017). The variations of the mechanical stress and strain have not been systematically evaluated and discussed for the arterial system under healthy and high blood pressure conditions either.

Clinical investigations measured the in vivo pulsatile expansion of the arterial wall and relative circumferential strain (Isnard et al. 1989; Boutouyrie et al. 1992), the results of which were used to guide in vitro cell stretching (Anwar et al. 2012). However, the residual stresses in vascular tissues compromise the strain measurement accuracy, so that the observed relative strain deviates substantially from the real strain with respect to the stress-free configuration (Chuong and Fung 1986). Nonlinear solid mechanics provides a theoretical way for quantifying these important mechanical factors onsite at the vascular wall (Holzapfel 2000; Humphrey 2002). By means of biaxial extension testing and histological evaluation, Kamenskiy et al. (2014) verified the hypothesis that the mechanical properties and microstructures of human arteries are dependent on specific locations and physiological functions. They estimated the material parameters and geometries of six major arteries, so that the in vivo stress/strain distributions are ready for detailed analysis. It has been reported that stress gradients within vascular tissues play important roles in cell movement (Lo et al. 2000), but their variations within three-dimensional vessel walls remain to be studied.

With the development of numerical methods for vascular mechanics, it is possible to predict the stress state in blood vessels more reliably with experimentally measured material parameters. But, over the past decades, main applications of these simulations are to develop artery-specific surgical therapies or medical devices for cardiovascular diseases (Gasser et al. 2002, 2010; Lally et al. 2005; Humphrey and Holzapfel 2012; Pierce et al. 2015), and the mechanical niches of vascular cells require more discussions. As an example, Sommer et al. (2018) recently investigated the biomechanical differences between subclavian and common iliac arteries under extension, inflation and torsion, which suggested that the high stiffness of the subclavian arteries explains the wall vulnerability during vascular operations. Their study focused on improving surgical interventions, rather than exploring the micromechanical environments for cells.

In this study, we tried to specify mechanical references of representative human vessels for the study of vascular cell mechano-biology. The contents are presented as follows. In Sect. 2, six human blood vessels including segments of the aortas and major branches are selected for stress analysis, before which the analytical methods are illustrated. In Sect. 3, the variations of the mechanical stress/strain distributions and stress gradients in the vessel walls under different blood pressures are illustrated. In Sect. 4, the biomechanical characteristics of blood vessels on cellular mechano-transduction are discussed, and a novel factor is proposed. Finally, conclusions are presented in Sect. 5.

2 Materials and methods

2.1 Blood vessel geometries and constitutive relations

Benefiting from the available mechanical experiments (Kamenskiy et al. 2014), six human blood vessels without severe diseases were selected, as illustrated in Fig. 1. They are thoracic (TA) and abdominal (AA) aortas, and common iliac (CIA), subclavian (SA), renal (RA), and common carotid (CCA) arteries. These blood vessels are representatives for the investigation of the mechanical environments in the vessel walls and for studying and comparing the characteristics of human blood vessels at different locations.

Fig. 1

Human aortas and major branches selected for stress analysis in this study. TA thoracic aorta, AA abdominal aorta, CIA common iliac artery, SA subclavian artery, RA renal artery, CCA common carotid artery

In this study, blood vessels were modeled as thick-wall straight tubes with residual stresses. An anisotropic constitutive formulation consisting of four exponential functions was used to describe the nonlinear mechanical behaviors of vessel walls (Kamenskiy et al. 2014):

$$W\left( {{\mathbf{C}},{\mathbf{M}}_{i} } \right) = \frac{{c_{0} }}{2}\left( {I_{1} - 3} \right) + \sum\limits_{i = 1}^{4} {\frac{{m_{i} c_{i1} }}{{4c_{i2} }}\left\{ {\exp \left[ {c_{i2} \left( {I_{i4} - 1} \right)^{2} } \right] - 1} \right\}} .$$

(1)

Here, $I_{1}$ is the first invariant of the right Cauchy–Green tensor C; ${\mathbf{M}}_{i} = {\mathbf{a}}_{i} \otimes {\mathbf{a}}_{i}$ is the tensor product of unit vector ${\mathbf{a}}_{i} = \left[ {\begin{array}{*{20}c} 0 & {\sin \gamma_{i} } & {\cos \gamma_{i} } \\ \end{array} } \right]^{\text{T}}$, i = 1, 2, 3, 4; $I_{i4} \left( {{\mathbf{C}},{\mathbf{a}}_{i} } \right) = {\mathbf{a}}_{i} \cdot {\mathbf{Ca}}_{i}$ is the pseudo-invariant of C and ${\mathbf{M}}_{i}$; $c_{0}$, $c_{i1}$, and $c_{i2}$ are material parameters with $c_{31} = c_{41}$ and $c_{32} = c_{42}$. In addition, the exponential terms in Eq. (1) only contribute when $I_{i4} > 1$ (Holzapfel et al. 2000), therefore:

$$m_{i} = \left\{ {\begin{array}{*{20}l} 1 \hfill & {{\text{when}}\;I_{i4} > 1} \hfill \\ 0 \hfill & {{\text{when}}\;I_{i4} \le 1} \hfill \\ \end{array} } \right. .$$

(2)

Referring to the work of Kamenskiy et al. (2014), the inner and outer radii in stress-free configuration, opening angle, and constitutive parameters for the six blood vessels are summarized in Table 1.

Table 1 The inner (R_i) and outer (R_o) radii in stress-free configuration, opening angle (α), and constitutive parameters for the selected blood vessels (Kamenskiy et al. 2014)

The opening angle α in Table 1 is used to quantify the residual stress in blood vessels, which is defined by two radii drawn from the midpoint of the inner wall to the tips of the inner wall (Fung 1993). It should be noted that the opening angle data are normally not available especially when the complicated three-dimensional geometries or multi-layer structures of blood vessels are considered. Some studies in the literature have reported to estimate the geometries and material parameters of vessel walls with non-invasively measured pressures and medical images (Masson et al. 2011; Wang et al. 2017). However, problems of non-uniqueness might arise when too many unknown parameters need to be fitted or the unknown parameters are sensitive to small changes of the in vivo measured data (Masson et al. 2011).

The axial in situ stretch $\lambda_{z}$ is used to quantify the magnitude of the axial loading, which is defined as the ratio of axial length of a vessel segment in situ to its ex situ axial length. As shown in Table 2, the $\lambda_{z}$ of AA, CIA, and CCA is collected from experimental measurements (Delfino et al. 1997; Schulze-Bauer et al. 2003; Holzapfel et al. 2007), and the $\lambda_{z}$ of TA is assumed to be the same as that of AA, and the $\lambda_{z}$ of SA and RA is not available in the experimental research and is assumed to be 1.1.

Table 2 The axial in situ stretch (λ_z) for the selected blood vessels

Hemodynamic studies demonstrate that the variation range of pulse blood pressure exhibits no obvious change along the human aorta (Boileau et al. 2015). Therefore, the six blood vessels in this study are assumed to experience the same systolic and diastolic pressures. According to Chobanian et al. (2003), the blood pressures adopted in the stress analysis are divided into five stages: normal blood pressure (P = 10.8 kPa), prehypertension (P = 13.2 kPa), stage 1 hypertension (P = 15.0 kPa), stage 2 hypertension (P = 16.7 kPa), and stage 3 hypertension (P = 17.7 kPa).

2.2 Analytical method

Based on nonlinear solid mechanics, the stress responses of the blood vessels are described, and the semi-analytical method of solving the equilibrium equations with proper boundary conditions is provided. More details are illustrated in Appendices 1 and 2.

3 Results

3.1 Effects of residual stress on stress distributions

Figure 2 illustrates the distributions of circumferential stresses $\sigma_{\theta \theta }$ across the vessel walls under normal blood pressure. In order to investigate the effects of residual stress on the stress distributions, the results with residual stress (w/RS) and those without residual stress (w/o RS) were compared. As shown in the figure, higher stress concentration is observed at the inner wall when the residual stress is not taken into account. The existence of the residual stress substantially alters the stress distributions in the vessel walls, resulting in a decrease in the $\sigma_{\theta \theta }$ at the inner wall and an increase in the $\sigma_{\theta \theta }$ at the outer wall.

Fig. 2

A comparison of stress distributions $\sigma_{\theta \theta }$ between the results with residual stress (w/RS) and those without residual stress (w/o RS) under normal blood pressure P = 10.8 kPa. The normalized radius $r_{\text{n}} = {{\left( {r - r_{\text{i}} } \right)} \mathord{\left/ {\vphantom {{\left( {r - r_{\text{i}} } \right)} {\left( {r_{\text{o}} - r_{\text{i}} } \right)}}} \right. \kern-0pt} {\left( {r_{\text{o}} - r_{\text{i}} } \right)}}$ is used as the abscissa, in which $r_{\text{n}} = 0$ denotes the inner radius $r_{\text{i}}$ and $r_{\text{n}} = 1$ denotes the outer radius $r_{\text{o}}$. The variation of $r_{\text{n}}$ from 0 to 1 represents the change of radial location from the inner wall to the outer wall

When the residual stress is taken into account, the stress concentrations at the inner walls of TA and AA are alleviated, however, the $\sigma_{\theta \theta }$ at the outer walls exhibits a significant increase by more than 170.0% (up to 100.0 kPa). For CIA, SA, and RA, the results of the $\sigma_{\theta \theta }$ along the radial direction with residual stress increase linearly. For CCA, the residual stress helps to form a uniform stress distribution across the vessel wall, and the magnitude of the $\sigma_{\theta \theta }$ is around 50.0 kPa.

3.2 Stress distributions

Figure 3 illustrates the distributions of circumferential stresses $\sigma_{\theta \theta }$ across the vessel walls at different hypertension stages. The stresses in TA, AA, and CIA exhibit an increase in the radial direction. The maximum stress occurring at the outer wall of AA exceeds 223.0 kPa at stage 3 hypertension. With the increasing blood pressure from normal to stage 3 hypertension, the stresses at the outer walls of TA and AA increase significantly by 92.4% and 95.4%, respectively. The stresses in SA and RA appear a linear variation through the wall thickness approximately. The increases in blood pressure raise the stresses in the walls of SA and RA to 77.3% and 73.1% on average, respectively. For CCA, a uniform stress distribution is observed in the vessel wall under normal blood pressure, while the stress decreases along the radial direction under hypertensions. The maximum stress at the inner wall of CCA increases by 117.8%, up to 111.0 kPa at stage 3 hypertension.

Fig. 3

Distributions of circumferential stresses $\sigma_{\theta \theta }$ through vessel walls. The blood pressure varies from normal (P = 10.8 kPa) to stage 3 hypertension (P = 17.7 kPa)

Considering that the stress values on the inner and outer surfaces of vessel walls are of particular interest for current vulnerable plaque research, the stresses of the inner, middle, and outer surfaces of the six blood vessels in Fig. 3 are summarized in Table 3 as a reference. The stress values of both normal pressure (10.8 kPa) and stage 3 hypertension (17.7 kPa) were listed, and their differences are shown in Table 3.

Table 3 The stresses (Unit: kPa) of the inner, middle, and outer surfaces of the selected blood vessels under normal blood pressure and hypertension

3.3 Strain distributions

Figure 4 illustrates the distributions of the circumferential strains $\varepsilon_{\theta \theta }$ across the vessel walls at different hypertension stages. The strains in TA, AA, and CIA exhibit a mild increase along the radial direction. The maximum strain around 0.41–0.46 is observed at the outer wall of CIA. The strains in SA and RA are around 0.3 and 0.2, respectively, which remain almost constant across the vessel walls. The strain in CCA decreases along the radial direction from the maximum value 0.33 at the inner wall.

Fig. 4

Distributions of circumferential strains $\varepsilon_{\theta \theta }$ through vessel walls. The blood pressure varies from normal (P = 10.8 kPa) to stage 3 hypertension (P = 17.7 kPa)

The strains of the inner, middle, and outer surfaces of the six blood vessels in Fig. 4 are summarized in Table 4.

Table 4 The strains of the inner, middle, and outer surfaces of the selected blood vessels under normal blood pressure and hypertension

3.4 Stress gradients

Based on the stress distributions in Fig. 3, the stress gradients ${{{\text{d}}\sigma_{\theta \theta } } \mathord{\left/ {\vphantom {{{\text{d}}\sigma_{\theta \theta } } {{\text{d}}r}}} \right. \kern-0pt} {{\text{d}}r}}$ are plotted against the normalized radius in Fig. 5. In comparison with TA and CIA, the stress gradient of AA increases obviously along the radial direction, which reaches the maximum of 311.6 kPa/mm at the outer wall. Blood pressure increase strongly affects the stress gradient at the outer wall of AA. The stress gradients in SA and RA decrease slightly along the radial direction, and all of them are less than 40.0 kPa/mm. The stress gradient in the wall of CCA is found to be negative which decreases rapidly near the inner wall and then approaches to a constant near the outer wall. A lowest value of − 91.5 kPa/mm is observed at the inner wall of CCA at stage 3 hypertension.

Fig. 5

Distributions of circumferential stress gradients ${{{\text{d}}\sigma_{\theta \theta } } \mathord{\left/ {\vphantom {{{\text{d}}\sigma_{\theta \theta } } {{\text{d}}r}}} \right. \kern-0pt} {{\text{d}}r}}$ through vessel walls. The blood pressure varies from normal (P = 10.8 kPa) to stage 3 hypertension (P = 17.7 kPa)

4 Discussions

Although lots of experiments for vascular cells in vitro have been reported in the studies of mechano-transduction, detailed examinations of the applied mechanical niches for cells, namely the stresses and strains comparing to their in vivo background, have not attracted sufficient attention yet. Benefiting from the progress in measuring the mechanical properties of human blood vessels (Kamenskiy et al. 2014), this study further evaluated three key mechanical factors, namely the mechanical stress, strain, and stress gradient, which constitute the mechanical niches of vascular cells and provide a site-specific mechanical reference for in vitro experiments.

4.1 “Site-specific” and “disease-specific” related biomechanical characteristics

In Fig. 3, three distinct characteristics of stress distribution were observed in the walls of the six selected blood vessels. In TA, AA, and CIA, the stresses increase along the radial direction. In contrast, the stresses decrease in CCA. In SA and RA, the stresses present relatively uniform distributions. Similar characteristics of strain distribution were also observed in the vessel walls, as shown in Fig. 4.

Higher strains on the inner surface of CCA may trigger the extracellular matrix (ECM) turnover (O’Callaghan and Williams 2000) and receptor-mediated apoptosis of the vascular smooth muscle cells (VSMCs) (Cattaruzza et al. 2000; Sotoudeh et al. 2002) and explain the inflammatory response initiated by the dysfunction of the endothelial cells (ECs) (Xiong et al. 2013; Spescha et al. 2014). In contrast, excessive stretch on the outer surface of TA, AA, and CIA may regulate the cytoskeletal remodeling and cell reorientation of fibroblasts (Yoshigi et al. 2005; Faust et al. 2011) and support the “outside-in” inflammation hypothesis initiated by the aortic adventitial fibroblasts (Maiellaro and Taylor 2007; Tieu et al. 2011; Michel et al. 2011; Libby and Hansson 2015).

4.2 Spatial gradient of mechanical stresses as a novel mechanical factor

As illustrated in Fig. 5, different stress gradients appeared in different vessel walls as the “site-specific” mechanical stresses and strains, which suggested that the stress gradient should be considered as a novel mechanical factor in mechano-biological studies.

It was found that relatively large stress gradient appeared at the outer surface of AA (311.6 kPa/mm), which was one order of magnitude higher than those of SA and RA. The increase in blood pressure significantly enhanced the stress gradient on the inner surface of CCA and the outer surface of TA and AA. Spatial gradients of mechanical stresses indicate that even slight dislocation of single vascular cells from their rest places could change the mechanical factors around them. As a consequence, the cell–cell and cell–ECM interactions may not only appreciate the local mechanical equilibrium of each cells (Maruthamuthu et al. 2011; Doyle et al. 2015), but also the physiological or pathological changes involving proliferation, differentiation or apoptosis of vascular cells that should be considered as dynamic processes coupled with cell behaviors such as migration, rather than static processes confined within isolated regions.

It should be noticed that three-dimensional matrices are of crucial importance in cell adhesion (Cukierman et al. 2001), which drive different cell behaviors versus two-dimensional environments in migration strategies and gene expression (Pedersen and Swartz 2005). From a mechanical point of view, three-dimensional matrices with proper spatial gradients of mechanical factors are helpful to rebuild the microenvironments of vascular tissues and improve the understanding of the dynamic coupling of cell biology with migration.

4.3 Current mechanical references for cell stretching in mechano-biological study

As illustrated in Fig. 6, the mechanical stretches for cell treatment reported in the literature were compared with the strains predicted in the six blood vessels. The strains at the inner, middle, and outer regions of the vessel walls were illustrated to characterize the strains for ECs, VSMCs, and fibroblasts, respectively. The mechanical strains in this study varied within the range of 0.17 to 0.46, but the applied mechanical stretches documented in the literature were mainly within the range of 0.05 to 0.25 for different types of cells, such as for ECs (Awolesi et al. 1995; Pourati et al. 1998; Qiu and Tarbell 2000; Cattaruzza et al. 2000; Peng et al. 2000; Wung et al. 2001; Wang et al. 2001; Shi et al. 2007; Berrout et al. 2012; Hu et al. 2013; Meza et al. 2016), for VSMCs (Li et al. 1998; Hipper and Isenberg 2000; Chapman et al. 2000; Cattaruzza et al. 2000, 2004; O’Callaghan and Williams 2000; Sotoudeh et al. 2002; Wernig et al. 2003; Sharifpoor et al. 2011; Wanjare et al. 2015), for fibroblasts (Yoshigi et al. 2005; Jungbauer et al. 2008; Faust et al. 2011; Livne et al. 2014; Cui et al. 2015), and for other cells (Wirtz and Dobbs 1990; Gwak et al. 2008; Heo and Lee 2011; Kim et al. 2016; Qiu et al. 2016). In some experiments, mechanical stretches were far outside the physiological range (≥ 1.2) for vascular cells (Yamada et al. 2000; Xiong et al. 2013).

Fig. 6

Mechanical strains on the inner, middle, and outer surfaces of the vessel walls for the six blood vessels, which are correlated with the strains of cell stretching in the literature. ECs (square), VSMCs (circle), fibroblasts (triangle), and other cells (star) are frequently used in experiments. The blood pressure varies from normal (P = 10.8 kPa) to stage 3 hypertension (P = 17.7 kPa)

The predicted stresses and strains in this study have considered the effect of residual stresses by assuming the opened-up sectors of the blood vessels as the stress-free configuration. The extra stresses or strains reserved by the residual stresses of blood vessels are often excluded in experiments for cells in vitro, so as the relative strains measured clinically. In order to interpret cellular responses more closely to their in vivo conditions, the absolute strains as presented in this work should be referenced.

5 Conclusions

In this study, we investigated the mechanical environments of vascular cells in six selected human blood vessels in order to illustrate the differences of reported studies of vascular cell mechano-transductions with in vivo mechanical niches of vascular cells and explore novel mechanical factors for in vitro cellular experiments. The results suggest that “disease-specific” dysfunctions of vascular cells should be investigated according to the “site-specific” mechanical environments. In addition to the mechanical stresses and strains, their spatial gradients are important mechanical factors when studying force-regulated cell behaviors. Exploring “factor-specific” mechano-transductions is one prerequisite for the development of vascular cell mechano-biology and tissue engineering.

Appendices

Appendix 1: Stress responses of blood vessels

It is well known that residual stress exists in the unloaded segment of blood vessels (Vaishnav and Vossoughi 1983; Chuong and Fung 1983). According to (Fung and Liu 1989; Han and Fung 1996), the segment can open up into a (nearly) stress-free sector when radial cutting occurs. Figure 7 illustrates three blood vessel configurations in cylindrical coordinates.

Fig. 7

Three blood vessel configurations in cylindrical coordinates: stress-free ($\varOmega_{0}$) with opening angle α, unloaded ($\varOmega_{ 1}$), and loaded ($\varOmega_{ 2}$) with applied blood pressure $P_{\text{i}}$. The opening angle α is defined by two radii drawn from the midpoint of the inner wall to its tips with $\varTheta_{0} = \pi - \alpha$

We refer to the stress-free state $\left( {\varOmega_{0} } \right)$ of Fig. 7 as the reference configuration. The position ${\mathbf{x}}_{1} \in \left( {\rho ,\vartheta ,\zeta } \right)$ of a material particle in the unloaded configuration $\left( {\varOmega_{1} } \right)$ is

$$\rho = \rho (R),\;\vartheta = \frac{\pi }{{\varTheta_{0} }}\varTheta ,\;\zeta = \varLambda Z$$

(3)

where $\varTheta_{0} = \pi - \alpha$, the axial stretch ratio $\varLambda { = 1} . 0$ for the assumption of plane strain (Han and Fung 1996).

The deformation gradient of the mapping from the stress-free to unloaded configuration is expressed as

$${\mathbf{F}}_{1} = \left[ {\begin{array}{*{20}l} {\frac{\partial \rho }{\partial R}} \hfill & {\frac{1}{R}\frac{\partial \rho }{\partial \varTheta }} \hfill & {\frac{\partial \rho }{\partial Z}} \hfill \\ {\rho \frac{\partial \vartheta }{\partial R}} \hfill & {\frac{\rho }{R}\frac{\partial \vartheta }{\partial \varTheta }} \hfill & {\rho \frac{\partial \vartheta }{\partial Z}} \hfill \\ {\frac{\partial \zeta }{\partial R}} \hfill & {\frac{1}{R}\frac{\partial \zeta }{\partial \varTheta }} \hfill & {\frac{\partial \zeta }{\partial Z}} \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {\frac{\partial \rho }{\partial R}} \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\frac{\pi \rho }{{\varTheta_{0} R}}} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right] .$$

(4)

Similarly, considering the mapping from the unloaded to loaded configuration, namely $r = r\left( \rho \right)$, $\theta = \vartheta + \beta \zeta$, $z = \lambda_{z} \zeta$, the associated deformation gradient is formulated as

$${\mathbf{F}}_{2} = \left[ {\begin{array}{*{20}c} {\frac{\partial r}{\partial \rho }} & {\frac{1}{\rho }\frac{\partial r}{\partial \vartheta }} & {\frac{\partial r}{\partial \zeta }} \\ {r\frac{\partial \theta }{\partial \rho }} & {\frac{r}{\rho }\frac{\partial \theta }{\partial \vartheta }} & {r\frac{\partial \theta }{\partial \zeta }} \\ {\frac{\partial z}{\partial \rho }} & {\frac{1}{\rho }\frac{\partial z}{\partial \vartheta }} & {\frac{\partial z}{\partial \zeta }} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\frac{\partial r}{\partial \rho }} & 0 & 0 \\ 0 & {\frac{r}{\rho }} & {r\beta } \\ 0 & 0 & {\lambda_{z} } \\ \end{array} } \right] ,$$

(5)

where β is the twist ratio and $\lambda_{z}$ is the axial stretch induced by the in vivo axial loading. As the torsion of the blood vessels is not investigated here, β = 0.

With respect to the reference (stress-free) configuration, the total deformation gradient for a loaded blood vessel is

$${\mathbf{F}} = {\mathbf{F}}_{2} \cdot {\mathbf{F}}_{1} = \left[ {\begin{array}{*{20}l} {\frac{\partial r}{\partial R}} \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\frac{\pi r}{{\varTheta_{0} R}}} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\lambda_{z} } \hfill \\ \end{array} } \right].$$

(6)

The right and left Cauchy–Green tensors are often used to measure the deformation and are defined as ${\mathbf{C}} = {\mathbf{F}}^{\text{T}} \cdot {\mathbf{F}}$ and ${\mathbf{b}} = {\mathbf{F}} \cdot {\mathbf{F}}^{\text{T}}$, respectively.

In order to describe the incompressibility of the vascular tissues, the constitutive relation $W\left( {{\mathbf{C}},{\mathbf{M}}_{i} } \right)$ in Eq. (1) is modified as follows:

$$\psi = W\left( {{\mathbf{C}},{\mathbf{M}}_{i} } \right) - \frac{1}{2}p\left( {I_{3} - 1} \right) ,$$

(7)

where the scalar p is the Lagrange multiplier and $I_{3}$ is the third invariant of the right Cauchy–Green tensor C with the constraint $I_{3} = 1$.

Using the chain rule, the second Piola–Kirchhoff stress tensor S is expressed as

$${\mathbf{S}} = 2\frac{\partial \psi }{{\partial {\mathbf{C}}}} = 2\left( {\frac{\partial W}{{\partial I_{1} }}\frac{{\partial I_{1} }}{{\partial {\mathbf{C}}}} + \sum\limits_{i = 1}^{4} {\frac{\partial W}{{\partial I_{i4} }}\frac{{\partial I_{i4} }}{{\partial {\mathbf{C}}}}} } \right) - p\frac{{\partial I_{3} }}{{\partial {\mathbf{C}}}} .$$

(8)

The derivatives of $I_{1}$, $I_{3}$, and $I_{i4}$ with respect to C have the following forms

$$\frac{{\partial I_{1} }}{{\partial {\mathbf{C}}}} = \frac{{\partial {\text{tr}}{\mathbf{C}}}}{{\partial {\mathbf{C}}}} = {\mathbf{I}},\;\frac{{\partial I_{3} }}{{\partial {\mathbf{C}}}} = \frac{{\partial { \det }{\mathbf{C}}}}{{\partial {\mathbf{C}}}} = I_{3} {\mathbf{C}}^{ - 1} ,\;\frac{{\partial I_{i4} }}{{\partial {\mathbf{C}}}} = \frac{{\partial \left( {{\mathbf{a}}_{i} \cdot {\mathbf{Ca}}_{i} } \right)}}{{\partial {\mathbf{C}}}} = {\mathbf{M}}_{i}$$

(9)

in which I is the identity matrix.

Substituting Eq. (9) into (8) results in

$${\mathbf{S}} = 2\left( {W_{1} {\mathbf{I}} + \sum\limits_{i = 1}^{4} {W_{i4} {\mathbf{M}}_{i} } } \right) - p{\mathbf{C}}^{ - 1} ,$$

(10)

with $W_{1} = {{\partial W} \mathord{\left/ {\vphantom {{\partial W} {\partial I_{1} }}} \right. \kern-0pt} {\partial I_{1} }},\;W_{i4} = {{\partial W} \mathord{\left/ {\vphantom {{\partial W} {\partial I_{i4} }}} \right. \kern-0pt} {\partial I_{i4} }}$.

The Piola transformation of Eq. (10) provides the Cauchy stress tensor as

$${\varvec{\upsigma}} = J^{ - 1} {\mathbf{FSF}}^{\text{T}} = {\varvec{\upsigma}}^{ * } - p{\mathbf{I}}$$

(11)

with ${\varvec{\upsigma}}^{ * } = 2\left( {W_{1} {\mathbf{b}} + \sum\nolimits_{i = 1}^{4} {W_{i4} {\mathbf{FM}}_{i} {\mathbf{F}}^{\text{T}} } } \right)$ for the sake of convenience.

Appendix 2: Solving equilibrium equations

Regardless of the body forces, the equilibrium equations can be expressed as

$${\text{div }}{\varvec{\upsigma}} = 0 .$$

(12)

The non-trivial component of Eq. (12) in cylindrical coordinates is

$$\frac{{{\text{d}}\sigma_{rr} }}{{{\text{d}}r}} + \frac{1}{r}\left( {\sigma_{rr} - \sigma_{\theta \theta } } \right) = 0 .$$

(13)

Neglecting the pressure on the outer wall ($P_{\text{o}} = 0$) and considering the internal blood pressure $P_{\text{i}}$, the boundary conditions are

$$\sigma_{rr} \left( {r_{\text{i}} } \right) = - P_{\text{i}} ,\;\sigma_{rr} \left( {r_{\text{o}} } \right) = 0,$$

(14)

where $r_{\text{i}}$ and $r_{\text{o}}$ are the inner and outer radii, respectively.

The integration of Eq. (13) from $r_{\text{i}}$ to r yields

$$\sigma_{rr} \left( r \right) - \sigma_{rr} \left( {r_{\text{i}} } \right) = \int\limits_{{r_{\text{i}} }}^{r} {\frac{1}{r}\left( {\sigma_{\theta \theta } - \sigma_{rr} } \right){\text{d}}r} .$$

(15)

Incorporating the stress components of Eq. (11) into (15) leads to the expression of the Lagrange multiplier:

$$p\left( r \right) = \sigma_{rr}^{ * } \left( r \right) + P_{\text{i}} - \int\limits_{{r_{\text{i}} }}^{r} {\frac{1}{r}\left( {\sigma_{\theta \theta }^{ * } \left( r \right) - \sigma_{rr}^{ * } \left( r \right)} \right){\text{d}}r} .$$

(16)

With the boundary conditions of Eq. (14), the integration of Eq. (13) from $r_{\text{i}}$ to $r_{\text{o}}$ relates the blood pressure $P_{\text{i}}$ to the vessel wall radial deformation; that is,

$$P_{\text{i}} = \int\limits_{{r_{\text{i}} }}^{{r_{\text{o}} }} {\frac{1}{r}\left( {\sigma_{\theta \theta }^{ * } \left( r \right) - \sigma_{rr}^{ * } \left( r \right)} \right){\text{d}}r} .$$

(17)

With Eq. (6), the incompressibility requires $\det {\mathbf{F}} \equiv 1$ and leads to

$${{\partial r} \mathord{\left/ {\vphantom {{\partial r} {\partial R}}} \right. \kern-0pt} {\partial R}} = {{\varTheta_{0} R} \mathord{\left/ {\vphantom {{\varTheta_{0} R} {\left( {\pi r\lambda_{z} } \right)}}} \right. \kern-0pt} {\left( {\pi r\lambda_{z} } \right)}} .$$

(18)

The integration of Eq. (18) from the inner to outer radius yields

$$r_{\text{o}}^{2} - r_{\text{i}}^{2} = \frac{{\varTheta_{0} }}{{\pi \lambda_{z} }}\left( {R_{\text{o}}^{2} - R_{\text{i}}^{2} } \right) .$$

(19)

By solving Eqs. (17) and (19) with the provided data of $R_{\text{i}}$, $R_{\text{o}}$, α, $P_{\text{i}}$, and $\lambda_{z}$, the inner ($r_{\text{i}}$) and outer ($r_{\text{o}}$) radii of the loaded configuration are obtained. Subsequently, the Lagrange multiplier $p\left( r \right)$ in Eq. (16) can be integrated numerically, following which the stress components in Eq. (11) are determined.