$$ \newcommand{\partd}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\partdd}[2]{\frac{\partial^{2} #1}{\partial {#2}^{2}}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\Int}{\int\limits} \newcommand{\D}{\displaystyle} \newcommand{\dA}{\; \mbox{dA}} \newcommand{\dz}{\; \mbox{dz}} $$
Blood flow in compliant vessels

 

 

 

Change of primary variables. The pQ, Au and other variations

As have been illustrated in the previous sections, there are several options with respect to selection of primary variables to represent the governing equations for wave propagation in compliant vessels. One might prefer to use the flow rate \( Q \), rather than the cross-sectionally averaged velocity \( v \), as such a formulation has a more natural mass conservation formulation. However, whether to choose cross-sectional area \( A \) or the pressure \( p \) is more a matter of taste, as long as a proper constitutive model represents their relation. It might therefore be beneficial to establish the relation between these two options of section of primary variables, keeping \( Q \) as the flow variable in both cases. Let the primary variable vectors be denoted by \( \mathbf{u}_p = [p, Q] \) and \( \mathbf{u}_A = [A, u] \), for \( A \) and \( p \) as primary variables, respectively. The corresponding, canonical differential equations \eqref{eq:vectorsystem} may the be presented: $$ \begin{align} \frac{\partial\mathbf{u}_p}{\partial t} + \frac{\partial\mathbf{M}_p}{\partial x} \; \frac{\partial\mathbf{u}_p}{\partial x} = 0 \tag{200} \\ \frac{\partial\mathbf{u}_A}{\partial t} + \frac{\partial\mathbf{M}_A}{\partial x} \; \frac{\partial\mathbf{u}_A}{\partial x} = 0 \tag{201} \end{align} $$

If we assume a functional relationship between \( \mathbf{u}_p = [p, Q] \) and \( \mathbf{u}_A = [A, u] \) their partial differentials are related by the Jacobian of \( \mathbf{u}_p = [p, Q] \) with respect to \( \mathbf{u}_A = [A, u] \): $$ \begin{equation} \tag{202} \frac{\partial \mathbf{u}_p}{\partial t} = \mathbf{J} \, \frac{\partial \mathbf{u}_A}{\partial t} \quad \text{ and } \quad \frac{\partial \mathbf{u}_p}{\partial x} = \mathbf{J} \, \frac{\partial \mathbf{u}_A}{\partial x} \qquad \text{ where } \quad \mathbf{J} = \left [ \begin{array}{cc} \displaystyle \frac{\partial p}{\partial A} & \displaystyle \frac{\partial p}{\partial u} \\ \displaystyle \frac{\partial Q}{\partial A} & \displaystyle \frac{\partial Q}{\partial u} \end{array} \right ] \end{equation} $$ Substitution of (202) into (200) yields after subsequent multiplication of the Jacobian inverse: $$ \begin{equation} \tag{203} \frac{\partial \mathbf{u}_A}{\partial t} +\mathbf{J}^{-1} \mathbf{M}_p \mathbf{J} \;\frac{\partial \mathbf{u}_A}{\partial x} = \frac{\partial \mathbf{u}_A}{\partial t} + \mathbf{M}_A \;\frac{\partial \mathbf{u}_A}{\partial x} = 0 \end{equation} $$ By comparing (203) with (201) we may relate \( \mathbf{M}_p \) and \( \mathbf{M}_A \): $$ \begin{equation} \tag{204} \mathbf{M}_A= \mathbf{J}^{-1} \mathbf{M}_p \mathbf{J} \end{equation} $$

From the relation in (204) it is easy to show that their eigenvalues of \( \mathbf{M}_p \) and \( \mathbf{M}_A \) are the same and the left \( \mathbf{L} \) and right \( \mathbf{R} \) eigenmatrices from the two systems are related with: $$ \begin{equation} \mathbf{R}_p = \mathbf{J} \, \mathbf{R}_A \quad \text{ and } \quad \mathbf{L}_p = \mathbf{L}_A \, \mathbf{J}^{-1} \tag{205} \end{equation} $$

Example: From the pQ-system to the Au-system

In this system we assume: $$ \begin{equation} p = p_0 + \frac{\partial p}{\partial A} \, (A - A_0) \quad \text{ and } \quad Q = u A \tag{206} \end{equation} $$ which results in the following Jacobian: $$ \begin{equation} \mathbf{J} = \left [ \begin{array}{cc} \displaystyle \frac{\partial p}{\partial A} & \displaystyle \frac{\partial p}{\partial u} \\ \displaystyle \frac{\partial Q}{\partial A} & \displaystyle \frac{\partial Q}{\partial u} \end{array} \right ] = \left [ \begin{array}{cc} \displaystyle \frac{1}{C} & 0\\ u & A \end{array} \right ] \tag{207} \end{equation} $$

The Riemann-invariants are approximated by: $$ \begin{equation} \Delta \omega= \mathbf{L}_p \, \Delta \mathbf{u}_p \tag{208} \end{equation} $$ and for small changes one may approximate: $$ \begin{equation} \Delta \mathbf{u}_p = \mathbf{J} \Delta \mathbf{u}_p \tag{209} \end{equation} $$ Substitution of (209) and (205) into (208): $$ \begin{equation} \Delta \omega= \mathbf{L}_p \, \Delta \mathbf{u}_p = \mathbf{L}_A \, \Delta \mathbf{u}_A \label{} \end{equation} $$