Annals of Biomedical Engineering

, Volume 44, Issue 5, pp 1553–1565 | Cite as

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

  • M. Keith SharpEmail author
  • Alexandra K. Diem
  • Roy O. Weller
  • Roxana O. Carare


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.


Brain Periarterial lymphatic flow Peristaltic flow Basement membrane Alzheimer’s disease Amyloid-β Darcy–Brinkman 


Dimensional Variables


Mean half height of channel


Cylinder radius


Wave amplitude


Wave speed


Drag force on a cylinder per unit length


Gravity vector


Half height of channel


Cylinder drag coefficient


Half distance between cylinders


Length of channel




Flow rate per unit breadth


Vessel radius



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

x direction velocity in the wave frame


Average velocity in the laboratory frame

V = (u,v,w)

Cartesian fluid velocity vector in the laboratory frame

x, y

Streamwise and radial coordinates


Wave length


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


Fluid density


Dynamic viscosity



Initial condition (when H = 1)




Over the length of the channel


Maximum volume fraction of cylinders (cylinders touch each other)

N, T

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


Over one wavelength

x, y,η, τ, ξ

Differentiation with respect to these variables

Dimensionless 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


Dimensionless flow rate in wave frame


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

Dimensionless Parameters

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

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



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

Amplitude ratio

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

Darcy number


Angle between the cylinder and the channel axis


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Copyright information

© Biomedical Engineering Society 2015

Authors and Affiliations

  • M. Keith Sharp
    • 1
    Email author
  • Alexandra K. Diem
    • 2
  • Roy O. Weller
    • 3
  • Roxana O. Carare
    • 4
  1. 1.Biofluid Mechanics Laboratory, Department of Mechanical EngineeringUniversity of LouisvilleLouisvilleUSA
  2. 2.Institute for Complex Systems Simulation and Computational Engineering and Design, Faculty of Engineering and the EnvironmentUniversity of SouthamptonSouthamptonUK
  3. 3.Faculty of Medicine, Southampton General HospitalUniversity of SouthamptonSouthampton, HampshireUK
  4. 4.Institute for Life Sciences, Southampton General HospitalUniversity of SouthamptonSouthamptonUK

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