The Flow Field along the Entire Length of Mouse Aorta and Primary Branches

Abstract

There is a spatial disposition to atherosclerosis along the aorta corresponding to regions of flow disturbances. The objective of the present study is to investigate the detailed distribution of hemodynamic parameters (wall shear stress (WSS), spatial gradient of wall shear stress (WSSG), and oscillatory shear index (OSI)) in the entire length of C57BL/6 mouse aorta with all primary branches (from ascending aorta to common iliac bifurcation). The detailed geometrical parameters (e.g., diameter and length of the vessels) were obtained from casts of entire aorta and primary branches of mice. The flow velocity was measured at the inlet of ascending aorta using Doppler flowprobe in mice. The outlet pressure boundary condition was estimated based on scaling law. The continuity and Navier–Stokes equations were solved using three-dimensional finite element method (FEM). The model prediction was tested by comparing the computed flow rate with the flow rate measured just before the common iliac bifurcation, and good agreement was found. It was also found that complex flow patterns occur at bifurcations between main trunk and branches. The major branches of terminal aorta, with the highest proportion of atherosclerosis, have the lowest WSS, and the relatively atherosclerotic-prone aortic arch has much more complex WSS distribution and higher OSI value than other sites. The low WSS coincides with the high OSI, which approximately obeys a power law relationship. Furthermore, the scaling law between flow and diameter holds in the entire aorta and primary branches of mice under pulsatile blood flow conditions. This model will eventually serve to elucidate the causal relation between hemodynamic patterns and atherogenesis in KO mice.

Appendix A

Hemodynamic Parameters

The Reynolds (Re), Womersley (α), and Dean (D _n ) numbers are defined, respectively, in main trunk of mouse aorta as follows:

$$ \text{Re} = \frac{{\rho V \cdot D}} {\mu } $$

(A1)

$$ \alpha = R{\sqrt {\frac{{\omega \rho }} {\mu }} } $$

(A2)

$$ D_{n} = {\left( {2\frac{R} {{R_{{{\text{curve}}}} }}} \right)}^{{1/2}} \cdot 4\text{Re} $$

(A3)

where $V = V_{{\min }} ,{\text{ }}V_{{\max }} ,{\text{ or }}V_{{{\text{mean}}}}$, R and D, ω, ρ, and μ represent minimum, maximum, or time-averaged velocity at the inlet of ascending aorta, radius and diameter of ascending aorta, angular frequency of beating hearts, blood mass density, and viscosity, respectively.

At any point of 3D FEM model, the stress can be represented as a nine-component tensor ($\ifmmode\expandafter\bar\else\expandafter\=\fi{\ifmmode\expandafter\bar\else\expandafter\=\fi{\tau }}$), which can be written as follows:

$$ \ifmmode\expandafter\bar\else\expandafter\=\fi{\ifmmode\expandafter\bar\else\expandafter\=\fi{\tau }} = {\left[ {\begin{array}{*{20}c} {{\tau _{{11}} }} & {{\tau _{{12}} }} & {{\tau _{{13}} }} \\ {{\tau _{{21}} }} & {{\tau _{{22}} }} & {{\tau _{{23}} }} \\ {{\tau _{{31}} }} & {{\tau _{{32}} }} & {{\tau _{{33}} }} \\ \end{array} } \right]} = 2\mu \overline{\overline{D}} = \mu {\left[ {\begin{array}{*{20}c} {{2\frac{{\partial u}} {{\partial x}}}} & {{\frac{{\partial u}} {{\partial y}} + \frac{{\partial v}} {{\partial x}}}} & {{\frac{{\partial u}} {{\partial z}} + \frac{{\partial w}} {{\partial x}}}} \\ {{\frac{{\partial u}} {{\partial y}} + \frac{{\partial v}} {{\partial x}}}} & {{2\frac{{\partial v}} {{\partial y}}}} & {{\frac{{\partial v}} {{\partial z}} + \frac{{\partial w}} {{\partial y}}}} \\ {{\frac{{\partial u}} {{\partial z}} + \frac{{\partial w}} {{\partial x}}}} & {{\frac{{\partial v}} {{\partial z}} + \frac{{\partial w}} {{\partial y}}}} & {{2\frac{{\partial w}} {{\partial z}}}} \\ \end{array} } \right]} $$

(A4)

where $\overline{\overline{D}} = 0.5 \cdot {[ {(\nabla {\mathbf{v}}) + (\nabla {\mathbf{v}})^{{\text{T}}} }]}$ is the shear rate tensor. The stress on the wall, its normal component, and it two tangential components can be written as, respectively:

$$ \ifmmode\expandafter\vec\else\expandafter\vec\fi{\tau } = \ifmmode\expandafter\bar\else\expandafter\=\fi{\ifmmode\expandafter\bar\else\expandafter\=\fi{\tau }} \cdot {\mathbf{n}},\quad \tau _{n} = {\mathbf{n}} \cdot \ifmmode\expandafter\bar\else\expandafter\=\fi{\ifmmode\expandafter\bar\else\expandafter\=\fi{\tau }} \cdot {\mathbf{n}},\quad \tau _{{t_{1} }} = {\mathbf{t}}_{{\text{1}}} \cdot \ifmmode\expandafter\bar\else\expandafter\=\fi{\ifmmode\expandafter\bar\else\expandafter\=\fi{\tau }} \cdot {\mathbf{n}}\;\;\;\;{\text{and}}\;\;\;\;\tau _{{t_{2} }} = {\mathbf{t}}_{{\text{2}}} \cdot \ifmmode\expandafter\bar\else\expandafter\=\fi{\ifmmode\expandafter\bar\else\expandafter\=\fi{\tau }} \cdot {\mathbf{n}} $$

(A5)

where n, t ₁ , and t ₂ are the unit vector in the normal and two tangential directions, respectively. The present time-averaged OSI can be written as follows:

$$ {\text{OSI}} = \frac{1} {2}{\left( {1 - \frac{{{\left| {\frac{1} {T}{\int_0^T {\ifmmode\expandafter\vec\else\expandafter\vec\fi{\tau }} }} \right|}}} {{\frac{1} {T}{\int_0^T {{\left| {\ifmmode\expandafter\vec\else\expandafter\vec\fi{\tau }} \right|}} }}}} \right)} $$

(A6)

The spatial derivatives of the stress can be obtained as follows:

$$ \nabla \ifmmode\expandafter\vec\else\expandafter\vec\fi{\tau } = {\left[ {\begin{array}{*{20}c} {{\frac{{\partial \tau _{n} }} {{\partial n}}}} & {{\frac{{\partial \tau _{n} }} {{\partial t_{1} }}}} & {{\frac{{\partial \tau _{n} }} {{\partial t_{2} }}}} \\ {{\frac{{\partial \tau _{{t_{1} }} }} {{\partial n}}}} & {{\frac{{\partial \tau _{{t_{1} }} }} {{\partial t_{1} }}}} & {{\frac{{\partial \tau _{{t_{1} }} }} {{\partial t_{2} }}}} \\ {{\frac{{\partial \tau _{{t_{2} }} }} {{\partial n}}}} & {{\frac{{\partial \tau _{{t_{2} }} }} {{\partial t_{1} }}}} & {{\frac{{\partial \tau _{{t_{2} }} }} {{\partial t_{2} }}}} \\ \end{array} } \right]} $$

(A7)

where n, t ₁, and t ₂ are the natural coordinates. As define by Buchanan et al.,4 the diagonal components $\partial \tau _{t_{1}}/ \partial t_{1}$ and $\partial \tau _{t_{2}} / \partial t_{2}$ generate intracellular tension, which causes widening and shrinking of cellular gap. However, the diagonal component $\partial \tau _{n}/{\partial n}$ can cause endothelial cells rotation, which may destroy the endothelial function too. Therefore, WSSG is defined as follows:

$$ {\text{WSSG}} = {\left[ {{\left( {\frac{{\partial \tau _{n} }} {{\partial n}}} \right)} + {\left( {\frac{{\partial \tau _{{t_{1} }} }} {{\partial t_{1} }}} \right)} + {\left( {\frac{{\partial \tau _{{t_{2} }} }} {{\partial t_{2} }}} \right)}} \right]}^{{\frac{1} {2}}} $$

(A8)

The time-averaged WSSG can be written:

$$ \hbox{time-averaged WSSG} = \frac{1} {T}{\int_0^T {{\text{WSSG}} \cdot {\text{d}}t} } $$

(A9)

To plot the shear stress in the entire computational domain, WSS is determined as the product of viscosity (μ) and wall shear rate ($\ifmmode\expandafter\dot\else\expandafter\.\fi{\gamma }$), which is defined as:

$$ {\text{WSS}} = \mu \ifmmode\expandafter\dot\else\expandafter\.\fi{\gamma } = \mu {\left[ \begin{aligned}{} & 2{\left( {{\left( {\frac{{\partial u}} {{\partial x}}} \right)}^{2} + {\left( {\frac{{\partial v}} {{\partial y}}} \right)}^{2} + {\left( {\frac{{\partial w}} {{\partial z}}} \right)}^{2} } \right)} + {\left( {{\left( {\frac{{\partial u}} {{\partial y}}} \right)}^{2} + {\left( {\frac{{\partial v}} {{\partial x}}} \right)}^{2} } \right)} + \\ & {\left( {{\left( {\frac{{\partial v}} {{\partial z}}} \right)}^{2} + {\left( {\frac{{\partial w}} {{\partial y}}} \right)}^{2} } \right)} + {\left( {{\left( {\frac{{\partial w}} {{\partial x}}} \right)}^{2} + {\left( {\frac{{\partial u}} {{\partial z}}} \right)}^{2} } \right)} \\ \end{aligned} \right]} $$

(A10)

The time-averaged WSS can be written as follows:

$$ \hbox{time-averaged WSS} = \frac{1} {T}{\int_0^T {WSS \cdot {\text{d}}t} } $$

(A11)

Equations (A6) and (A8–A11) were used to calculate the OSI, WSSG, and WSS in the FEM model.