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

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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.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Numbers 11872040 and 11232010).

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Correspondence to Xiaobo Gong.

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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
figure 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.

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Yang, S., Gong, X., Qi, Y. et al. Comparative study of variations in mechanical stress and strain of human blood vessels: mechanical reference for vascular cell mechano-biology. Biomech Model Mechanobiol 19, 519–531 (2020). https://doi.org/10.1007/s10237-019-01226-1

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