Pulsations with reflected boundary waves: a hydrodynamic reverse transport mechanism for perivascular drainage in the brain

Abstract

Beta-amyloid accumulation within arterial walls in cerebral amyloid angiopathy is associated with the onset of Alzheimer’s disease. However, the mechanism of beta-amyloid clearance along peri-arterial pathways in the brain is not well understood. In this study, we investigate a transport mechanism in the arterial basement membrane consisting of forward-propagating waves and their reflections. The arterial basement membrane is modeled as a periodically deforming annulus filled with an incompressible single-phase Newtonian fluid. A reverse flow, which has been suggested in literature as a beta-amyloid clearance pathway, can be induced by the motion of reflected boundary waves along the annular walls. The wave amplitude and the volume of the annular region govern the flow magnitude and may have important implications for an aging brain. Magnitudes of transport obtained from control volume analysis and numerical solutions of the Navier–Stokes equations are presented.

Appendices

Appendix 1: Momentum correction factor for an annulus

The dimensionless momentum-flux correction factor at a control surface k is

$$\begin{aligned} \beta _k =\frac{1}{A_k }\int \limits _{A_k}{\left( {\frac{u_k }{\bar{{u}}_k }} \right) ^{2}dA_k} \end{aligned}$$

(13)

where $u_{k}$ is the velocity profile at the control surface. $\beta _{k}=1$ represents a uniform flow profile while a parabolic flow profile leads to $\beta _{k}> 1$. The velocity profile for a steady, incompressible, laminar, unidirectional flow in an annulus with no-slip boundary conditions is

$$\begin{aligned} u_k (r)=\frac{r_{i,k}^{2}-r_{o,k} ^{2}}{4\mu }\frac{dp}{dx}\left( {\frac{r^{2}-r_{o,k}^{2}}{r_{i,k} ^{2}-r_{o,k} ^{2}}-\frac{\ln \left( {\frac{r}{r_{o,k}}} \right) }{\ln \left( {\frac{r_{i,k} }{r_{o,k} }} \right) }}\right) \end{aligned}$$

(14)

where dp/dx is the local pressure gradient and $\mu$ is the fluid viscosity, and

$$\begin{aligned} \bar{{u}}_{k}= & {} -\frac{1}{2\mu }\frac{dp}{dx}\int \limits _{r_{i,k} }^{r_{o,k} } {\left( {\frac{r^{2}-r_{o,k}^{2}}{r_{i,k} ^{2}-r_{o,k} ^{2}}-\frac{\ln \left( {\frac{r}{r_{o,k} }} \right) }{\ln \left( {\frac{r_{i,k} }{r_{o,k} }} \right) }} \right) } rdr\nonumber \\= & {} -\frac{1}{8\mu }\frac{dp}{dx}\left( {r_{i,k}^{2}+r_{o,k} ^{2}-\frac{r_{i,k} ^{2}-r_{o,k} ^{2}}{\ln \left( {\frac{r_{i,k} }{r_{o,k} }} \right) }} \right) . \end{aligned}$$

(15)

Appendix 2: Derivation of the control volume flow rate formula

Equation (8) can be re-written in terms of $\bar{{u}}_{2}$ and $\alpha$,

$$\begin{aligned} \bar{{u}}_2 =\pm \bar{{u}}_1 \left( {\frac{1}{\alpha }} \right) ^{\frac{1}{2}}\left( {1-\frac{1}{\bar{{u}}_1 ^{2}\beta _1 A_1 }\frac{dP}{dt}} \right) ^{\frac{1}{2}}, \end{aligned}$$

(16)

where $\bar{u}_{2}$ must remain real at all times and requires ${dP}/{dt}<\bar{{u}}_{1}^{2}\beta _{1} A_1$. The periodic nature of the rate of change of momentum in the control volume allows us to infer that $\left| {{dP}/{dt}} \right| <\bar{{u}}_1 ^{2}\beta _1 A_1$. We thus employ a binomial expansion of Eq. (16),

$$\begin{aligned} \bar{{u}}_2 =\pm \bar{{u}}_1 \left( {\frac{1}{\alpha }} \right) ^{1/2}\left[ {1-\frac{1}{2}\left( {\frac{1}{\bar{{u}}_1 ^{2}\beta _1 A_1}\frac{dP}{dt}} \right) -\frac{1}{8}\left( {\frac{1}{\bar{{u}}_1 ^{2}\beta _1 A_1 }\frac{dP}{dt}} \right) ^{2}+\cdots }\right] .\qquad \end{aligned}$$

(17)

Since at any given instant of time a velocity must be unique, we determine the sign in Eq. (17) by considering the leading order term and substitute it into Eq. (7), which leads to

$$\begin{aligned} \bar{{u}}_1 =\frac{dV}{dt}\frac{1}{A_1 }\frac{1}{1\mp \alpha ^{1/2}}. \end{aligned}$$

(18)

The minus sign solution in Eq. (18) is physically unreasonable because during each deformation cycle, there must exist instances when $\alpha =1$. Thus, at those instances

$$\begin{aligned} \mathop {\lim }\limits _{\alpha =1} \frac{1}{1-\alpha ^{1/2}}\rightarrow \infty \end{aligned}$$

which leads to singularity for $\bar{{u}}_1$. This indicates that the only plausible sign for Eq. (17) must be negative and

$$\begin{aligned} \bar{{u}}_2 =-\bar{{u}}_1 \left( {\frac{1}{\alpha }} \right) ^{1/2}\left[ {1-\frac{1}{2}\left( {\frac{1}{\bar{{u}}_1 ^{2}\beta _1 A_1 }\frac{dP}{dt}} \right) -\frac{1}{8}\left( {\frac{1}{\bar{{u}}_1 ^{2}\beta _1 A_1 }\frac{dP}{dt}} \right) ^{2}+\cdots } \right] .\qquad \end{aligned}$$

(19)

When the two leading order terms in Eq. (19) are considered and substituted into Eq. (7), it yields a quadratic equation in terms of $\bar{{u}}_1$,

$$\begin{aligned} \left( {1+\alpha ^{1/2}} \right) A_1 \bar{{u}}_1 ^{2}-\left( {\frac{dV}{dt}} \right) \bar{{u}}_1 -\frac{1}{2\beta _1 }\alpha ^{1/2}\frac{dP}{dt}=0, \end{aligned}$$

(20)

whose solution is

$$\begin{aligned} \bar{{u}}_1 =\frac{\left( {\frac{dV}{dt}} \right) \pm \sqrt{\left( {\frac{dV}{dt}} \right) ^{2}-2\frac{A_1 }{\beta _1 }\frac{dP}{dt}\left( {1+\alpha ^{1/2}} \right) }}{2\left( {1+\alpha ^{1/2}} \right) A_1 }. \end{aligned}$$

(21)

Again, $\bar{{u}}_1$ must remain real at all times, and given the periodic nature of dP/dt, we can infer that

$$\begin{aligned} \left| {2\frac{A_1 }{\beta _1 }\left( {\frac{dV}{dt}} \right) ^{-2}\frac{dP}{dt}\left( {1+\alpha ^{1/2}} \right) } \right| \le 1. \end{aligned}$$

(22)

Once again by invoking the binomial series expansion of the square root term in Eq. (21) and keeping the two leading order terms, we get

$$\begin{aligned} \bar{{u}}_1 \approx \frac{\left( {\frac{dV}{dt}} \right) \pm \left( {\frac{dV}{dt}} \right) \left[ {1+\frac{A_1 }{\beta _1 }\frac{dP}{dt}\left( {\frac{dV}{dt}} \right) ^{-2}\left( {\alpha +\alpha ^{1/2}} \right) } \right] }{2\left( {1+\alpha ^{1/2}} \right) A_1 }. \end{aligned}$$

(23)

Numerical evaluation of Eq. (23) requires us to obtain an approximate expression for dP/dt. We define a spatially averaged velocity in the x-direction as $u_{avg,S}=(1/V) \int \int \int (\mathbf{u}\cdot \hat{{x}})dV = P/V$ where u is velocity field vector and $\hat{{x}}$ is the unit direction vector pointing in the axial direction. Since the flow is periodic, a velocity averaged over a space of one wavelength is approximately equal to a timed-averaged velocity $u_{avg,T}$ at $A_{1}$ over one period, or

$$\begin{aligned} u_{avg,S} \approx u_{avg,T} =\frac{1}{\tau }\int _{0}^{\tau } {\bar{{u}}_1 \left( t \right) dt} =\frac{Q_{\textit{CV}} }{\int _{0}^{\tau } {A_1 \left( t \right) dt}}. \end{aligned}$$

(24)

Given that $u_{avg,T}$ is not a function of time,

$$\begin{aligned} \frac{dP}{dt}=\frac{d}{dt}\left( {u_{avg,S} V} \right) \approx \frac{d}{dt}\left( {u_{avg,T} V} \right) =\frac{Q_{\textit{CV}} }{\int _{0}^{\tau } {A_1 \left( t \right) dt} }\frac{dV}{dt}. \end{aligned}$$

(25)

Substituting Eq. (25) into Eq. (23) yields

$$\begin{aligned}&Q_{\textit{CV}} \approx \int _{0}^{\tau } {\frac{dV}{dt}\frac{1}{2\left( {1+\alpha ^{1/2}} \right) }dt}\nonumber \\&\quad \qquad \pm \int _{0}^{\tau } {\frac{1}{2\left( {1+\alpha ^{1/2}} \right) }\left[ {\frac{dV}{dt}+\frac{A_1 }{\beta _1 }\frac{Q_{\textit{CV}} }{\int _{0}^{\tau } {A_1 \left( t \right) dt} }\left( {\alpha +\alpha ^{1/2}} \right) } \right] dt}. \end{aligned}$$

(26)

Equation (26) shows two possible solutions for $Q_{\textit{CV}}$, which also must be unique. We first consider Eq. (26) with the negative sign, which yields

$$\begin{aligned}&Q_{\textit{CV}} =-\int _{0}^{\tau } {\left[ {\frac{A_1 }{2\beta _1 \left( {1+\alpha ^{1/2}} \right) }\frac{Q_{\textit{CV}} \left( {\alpha +\alpha ^{1/2}} \right) }{\int _{0}^{\tau } {A_1 \left( t \right) dt} }} \right] dt}\nonumber \\&\qquad \quad =-\frac{Q_{\textit{CV}} }{2\beta _1 \int _{0}^{\tau } {A_1 \left( t \right) dt} }\int _{0}^{\tau } {A_1 \alpha ^{1/2}dt}. \end{aligned}$$

(27)

Rearranging Eq. (27) leads to $Q_{\textit{CV}}=0$, regardless of the annular deformation, which is not physically plausible given that a periodically deforming control volume due to traveling boundary waves should result in non-zero overall flow. Choosing the positive sign in Eq. (26) thus leads to Eq. (7).