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Peristalsis with Oscillating Flow Resistance: A Mechanism for Periarterial Clearance of Amyloid Beta from the Brain

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Abstract

Alzheimer’s disease is characterized by accumulation of amyloid-β (Aβ) in the brain and in the walls of cerebral arteries. The focus of this work is on clearance of Aβ along artery walls, the failure of which may explain the accumulation of Aβ in Alzheimer’s disease. Periarterial basement membranes form continuous channels from cerebral capillaries to major arteries on the surface of the brain. Arterial pressure pulses drive peristaltic flow in the basement membranes in the same direction as blood flow. Here we forward the hypothesis that flexible structures within the basement membrane, if oriented such they present greater resistance to forward than retrograde flow, may cause net reverse flow, advecting Aβ along with it. A solution was obtained for peristaltic flow with low Reynolds number, long wavelength compared to channel height and small channel height compared to vessel radius in a Darcy–Brinkman medium representing a square array of cylinders. Results show that retrograde flow is promoted by high cylinder volume fraction and low peristaltic amplitude. A decrease in cylinder concentration and/or an increase in amplitude, both of which may occur during ageing, can reduce retrograde flow or even cause a transition from retrograde to forward flow. Such changes may explain the accumulation of Aβ in the brain and in artery walls in Alzheimer’s disease.

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Abbreviations

a :

Mean half height of channel

a c :

Cylinder radius

b :

Wave amplitude

c :

Wave speed

f :

Drag force on a cylinder per unit length

G :

Gravity vector

h :

Half height of channel

K :

Cylinder drag coefficient

l :

Half distance between cylinders

L :

Length of channel

p :

Pressure

q :

Flow rate per unit breadth

R :

Vessel radius

t :

Time

\(\hat{u} \equiv u - c\) :

x direction velocity in the wave frame

\(\bar{u}\) :

Average velocity in the laboratory frame

V = (u,v,w):

Cartesian fluid velocity vector in the laboratory frame

x, y :

Streamwise and radial coordinates

λ :

Wave length

\(\psi\) :

Stream function \(\psi_{x} \equiv u, \, \psi_{y} \equiv v\)

ρ :

Fluid density

μ :

Dynamic viscosity

0 :

Initial condition (when H = 1)

c :

Cylinder

L :

Over the length of the channel

max :

Maximum volume fraction of cylinders (cylinders touch each other)

N, T :

Normal and tangential (orientation of the cylinder relative to the flow)

λ :

Over one wavelength

x, y, η, τ, ξ :

Differentiation with respect to these variables

\(H \equiv \frac{{h\left( {x,t} \right)}}{a}\) :

Normalized channel half height

\(\bar{H} = 1\) :

Average channel half height over one wave period

\(P \equiv \frac{{2\pi a^{2} }}{\lambda \mu c}p\) :

Dimensionless pressure

\(Q \equiv \frac{1}{ac}q\) :

Dimensionless flow rate in the laboratory frame

\(\hat{Q}\) :

Dimensionless flow rate in wave frame

\(\bar{Q}\) :

Spatial mean dimensionless flow rate in the laboratory frame

\(U \equiv \frac{u}{c}\) :

Dimensionless x direction velocity

\(\chi \equiv \frac{\psi }{ac}\) :

Normalized stream function

\(\varepsilon = \frac{{\pi a_{c}^{2} }}{{4l^{2} }}\) :

Volume fraction of cylinders

\(\varPhi = 1 - \varepsilon\) :

Porosity of the channel

\(\eta \equiv \frac{y}{a}\) :

Normalized radial direction coordinate

\(\tau \equiv 2\pi \frac{ct}{\lambda }\) :

Normalized time

\(\xi \equiv 2\pi \frac{x}{\lambda }\) :

Normalized axial direction coordinate

\(Re \equiv \frac{\rho ac\alpha }{\mu }\) :

Reynolds number

\(\alpha \equiv \frac{2\pi a}{\lambda }\) :

Wave number (related to slope and curvature of channel)

\(\beta \equiv \frac{{a_{c} }}{a}\) :

Cylinder radius to channel half height ratio

κ :

Permeability

\(\phi \equiv \frac{b}{a}\) :

Amplitude ratio

\(\gamma = \frac{h\sqrt K }{2l}\) :

Darcy number

θ :

Angle between the cylinder and the channel axis

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Authors and Affiliations

  1. Biofluid Mechanics Laboratory, Department of Mechanical Engineering, University of Louisville, 200 Sackett Hall, Louisville, KY, 40292, USA

    M. Keith Sharp

  2. Institute for Complex Systems Simulation and Computational Engineering and Design, Faculty of Engineering and the Environment, University of Southampton, Building 25, Highfield, Southampton, SO17 1BJ, UK

    Alexandra K. Diem

  3. Faculty of Medicine, Southampton General Hospital, University of Southampton, Southampton, Hampshire, SO16 6YD, UK

    Roy O. Weller

  4. Institute for Life Sciences, Southampton General Hospital, University of Southampton, MP806, Clinical Neurosciences, Tremona Road, Southampton, SO16 6YD, UK

    Roxana O. Carare

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  1. M. Keith Sharp
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Correspondence to M. Keith Sharp.

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Associate Editor Aleksander S. Popel oversaw the review of this article.

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Sharp, M.K., Diem, A.K., Weller, R.O. et al. Peristalsis with Oscillating Flow Resistance: A Mechanism for Periarterial Clearance of Amyloid Beta from the Brain. Ann Biomed Eng 44, 1553–1565 (2016). https://doi.org/10.1007/s10439-015-1457-6

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