Waves in blood vessels

The flow of blood in our bodies is basically fluid flow in a network of pipes. Unlike rigid pipes, the walls in the blood vessels are elastic and will increase their diameter when the pressure rises. The elastic forces will then push the wall back and accelerate the fluid. This interaction between the flow of blood and the deformation of the vessel wall results in waves traveling along our blood vessels.

A model for one-dimensional waves along blood vessels can be derived from averaging the fluid flow over the cross section of the blood vessels. Let \(x\) be a coordinate along the blood vessel and assume that all cross sections are circular, though with different radii \(R(x,t)\). The main quantities to compute is the cross section area \(A(x,t)\), the averaged pressure \(P(x,t)\), and the total volume flux \(Q(x,t)\). The area of this cross section is

\[\tag{156} A(x,t) = 2\pi\int_{0}^{R(x,t)} rdr,\]

Let \(v_x(x,t)\) be the velocity of blood averaged over the cross section at point \(x\). The volume flux, being the total volume of blood passing a cross section per time unit, becomes

\[\tag{157} Q(x,t) = A(x,t)v_x(x,t) \thinspace\]

Mass balance and Newton’s second law lead to the PDEs

\[\tag{158} \frac{\partial A}{\partial t} + \frac{\partial Q}{\partial x} = 0,\]
\[\tag{159} \frac{\partial Q}{\partial t} + \frac{\gamma+2}{\gamma + 1} \frac{\partial}{\partial x}\left(\frac{Q^2}{A}\right) + \frac{A}{\varrho}\frac{\partial P}{\partial x} = -2\pi (\gamma + 2)\frac{\mu}{\varrho}\frac{Q}{A},\]

where \(\gamma\) is a parameter related to the velocity profile, \(\varrho\) is the density of blood, and \(\mu\) is the dynamic viscosity of blood.

We have three unknowns \(A\), \(Q\), and \(P\), and two equations (158) and (159). A third equation is needed to relate the flow to the deformations of the wall. A common form for this equation is

\[\tag{160} \frac{\partial P}{\partial t} + \frac{1}{C} \frac{\partial Q}{\partial x} =0,\]

where \(C\) is the compliance of the wall, given by the constitutive relation

\[\tag{161} C = \frac{\partial A}{\partial P} + \frac{\partial A}{\partial t},\]

which require a relationship between \(A\) and \(P\). One common model is to view the vessel wall, locally, as a thin elastic tube subject to an internal pressure. This gives the relation

\[P=P_0 + \frac{\pi h E}{(1-\nu^2)A_0}(\sqrt{A} - \sqrt{A_0}),\]

where \(P_0\) and \(A_0\) are corresponding reference values when the wall is not deformed, \(h\) is the thickness of the wall, and \(E\) and \(\nu\) are Young’s modulus and Poisson’s ratio of the elastic material in the wall. The derivative becomes

\[\tag{162} C = \frac{\partial A}{\partial P} = \frac{2(1-\nu^2)A_0}{\pi h E}\sqrt{A_0} + 2\left(\frac{(1-\nu^2)A_0}{\pi h E}\right)^2(P-P_0) {\thinspace .}\]

Another (nonlinear) deformation model of the wall, which has a better fit with experiments, is

\[P = P_0\exp{(\beta (A/A_0 - 1))},\]

where \(\beta\) is some parameter to be estimated. This law leads to

\[\tag{163} C = \frac{\partial A}{\partial P} = \frac{A_0}{\beta P} {\thinspace .}\]

Reduction to the standard wave equation. It is not uncommon to neglect the viscous term on the right-hand side of (159) and also the quadratic term with \(Q^2\) on the left-hand side. The reduced equations (159) and (160) form a first-order linear wave equation system:

\[\tag{164} C\frac{\partial P}{\partial t} = - \frac{\partial Q}{\partial x},\]
\[\tag{165} \frac{\partial Q}{\partial t} = - \frac{A}{\varrho}\frac{\partial P}{\partial x} {\thinspace .}\]

These can be combined into standard 1D wave equation PDE by differentiating the first equation with respect \(t\) and the second with respect to \(x\),

\[\frac{\partial}{\partial t}\left( C\frac{\partial P}{\partial t} \right) = \frac{\partial}{\partial x}\left( \frac{A}{\varrho}\frac{\partial P}{\partial x}\right),\]

which can be approximated by

\[\tag{166} \frac{\partial^2 Q}{\partial t^2} = c^2\frac{\partial^2 Q}{\partial x^2},\quad c = \sqrt{\frac{A}{\varrho C}},\]

where the \(A\) and \(C\) in the expression for \(c\) are taken as constant reference values.

Electromagnetic waves

Light and radio waves are governed by standard wave equations arising from Maxwell’s general equations. When there are no charges and no currents, as in a vacuum, Maxwell’s equations take the form

\[\begin{split}\nabla\cdot\pmb{E} &= 0,\\ \nabla\cdot\pmb{B} &= 0,\\ \nabla\times\pmb{E} &= -\frac{\partial\pmb{B}}{\partial t},\\ \nabla\times\pmb{B} &= \mu_0\epsilon_0\frac{\partial\pmb{E}}{\partial t},\end{split}\]

where \(\epsilon_0=8.854187817620\cdot 10^{-12}\) (F/m) is the permittivity of free space, also known as the electric constant, and \(\mu_0=1.2566370614\cdot 10^{-6}\) (H/m) is the permeability of free space, also known as the magnetic constant. Taking the curl of the two last equations and using the mathematical identity

\[\nabla\times (\nabla\times \pmb{E}) = \nabla(\nabla \cdot \pmb{E}) - \nabla^2\pmb{E} = - \nabla^2\pmb{E}\hbox{ when }\nabla\cdot\pmb{E}=0,\]

gives the wave equation governing the electric and magnetic field:

\[\tag{167} \frac{\partial^2\pmb{E}}{\partial t^2} = c^2\nabla^2\pmb{E},\]
\[\tag{168} \frac{\partial^2\pmb{B}}{\partial t^2} = c^2\nabla^2\pmb{B},\]

with \(c=1/\sqrt{\mu_0\epsilon_0}\) as the velocity of light. Each component of \(\pmb{E}\) and \(\pmb{B}\) fulfills a wave equation and can hence be solved independently.