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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 Sharp
  • Alexandra K. Diem
  • Roy O. Weller
  • Roxana O. Carare
Article

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.

Keywords

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

Nomenclature

Dimensional Variables

a

Mean half height of channel

ac

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

Subscripts

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

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

\(\hat{Q}\)

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

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

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

References

  1. 1.
    Abbott, N. J. Evidence for bulk flow of brain interstitial fluid: significance for physiology and pathology. Neurochem. Int. 45(4):545–552, 2004.CrossRefPubMedGoogle Scholar
  2. 2.
    Arbel-Ornath, M., E. Hudry, K. Eikermann-Haerter, S. Hou, J. L. Gregory, L. Zhao, R. A. Betensky, M. P. Frosch, S. M. Greenberg, and B. J. Bacskai. Interstitial fluid drainage is impaired in ischemic stroke and Alzheimer’s disease mouse models. Acta Neuropathol. 126:353–364, 2013. doi: 10.1007/s00401-013-1145-2.CrossRefPubMedGoogle Scholar
  3. 3.
    Bogren, H. G., R. H. Klipstein, D. N. Firmin, R. H. Mohiaddin, S. R. Underwood, S. O. Rees, and D. B. Longmore. Quantitation of antegrade and retrograde blood flow in the human aorta by magnetic resonance velocity mapping. Am. Heart J. 117(6):1214–1222, 1989.CrossRefPubMedGoogle Scholar
  4. 4.
    Bradbury, M. W., H. F. Cserr, and R. J. Westrop. Drainage of cerebral interstitial fluid into deep cervical lymph of the rabbit. Am. J. Physiol. 240(4):F329–F336, 1981.PubMedGoogle Scholar
  5. 5.
    Byron, A., M. J. Randles, J. D. Humphries, A. Mironov, H. Hamidi, S. Harris, P. W. Mathieson, M. A. Saleem, S. S. Satchell, R. Zent, M. J. Humphries, and R. Lennon. Glomerular cell cross-talk influences composition and assembly of extracellular matrix. J. Am. Soc. Nephrol. 2014. doi: 10.1681/ASN.2013070795.Google Scholar
  6. 6.
    Carare, R. O., M. Bernardes-Silva, T. A. Newman, A. M. Page, J. A. Nicoll, V. H. Perry, and R. O. Weller. Solutes, but not cells, drain from the brain parenchyma along basement membranes of capillaries and arteries: significance for cerebral amyloid angiopathy and neuroimmunology. Neuropathol. Appl. Neurobiol. 34(2):131–144, 2008.CrossRefPubMedGoogle Scholar
  7. 7.
    Carare, R. O., C. A. Hawkes, M. Jeffrey, R. N. Kalaria, and R. O. Weller. Cerebral amyloid angiopathy, Prion angiopathy, CADASIL and the spectrum of Protein Elimination-Failure Angiopathies (PEFA) in neurodegenerative disease with a focus on therapy. Neuropathol. Appl. Neurobiol. 39(6):593–611, 2013. doi: 10.1111/nan.12042.CrossRefPubMedGoogle Scholar
  8. 8.
    Ford, M. D., N. Alperin, S. H. Lee, D. W. Holdsworth, and D. A. Steinman. Characterization of volumetric flow rate waveforms in the normal internal carotid and vertebral arteries. Physiol. Meas. 26:477–488, 2005.CrossRefPubMedGoogle Scholar
  9. 9.
    Hawkes, C. A., W. Hartig, J. Kacza, R. Schliebs, R. O. Weller, J. A. Nicoll, and R. O. Carare. Perivascular drainage of solutes is impaired in the ageing mouse brain and in the presence of cerebral amyloid angiopathy. Acta Neuropathol. 121(4):431–443, 2011; (Epub 2011/01/25).CrossRefPubMedGoogle Scholar
  10. 10.
    Hawkes, C. A., P. M. Sullivan, S. Hands, R. O. Weller, J. A. Nicoll, and R. O. Carare. Disruption of arterial perivascular drainage of amyloid-β from the brains of mice expressing the human APOE ε4 allele. PLoS One 7(7):e41636, 2012. doi: 10.1371/journal.pone.0041636.CrossRefPubMedPubMedCentralGoogle Scholar
  11. 11.
    Hohenster, E., and P. D. Yurchenco. Laminins in basement membrane assembly. Cell Adhesion Migration 7(1):56–63, 2013.CrossRefGoogle Scholar
  12. 12.
    Iliff, J. J., M. Wang, Y. Liao, B. A. Plogg, W. Peng, G. A. Gundersen, H. Benveniste, G. E. Vates, R. Deane, S. A. Goldman, E. A. Nagelhus, and M. Nedergaard. A paravascular pathway facilitates CSF flow through the brain parenchyma and the clearance of interstitial solutes, including amyloid beta. Sci. Transl. Med. 4(147):147ra11, 2012; (Epub 2012/08/17).Google Scholar
  13. 13.
    Iliff, J. J., M. Wang, D. M. Zeppenfeld, A. Venkataraman, B. A. Plog, Y. Liao, R. Deane, and M. Nedergaard. Cerebral arterial pulsation drives paravascular CSF-interstitial fluid exchange in the murine brain. J. Neurosci. Off. J. Soc. Neurosci. 33:18190–18199, 2013.CrossRefGoogle Scholar
  14. 14.
    Jaffrin, M. Y., and A. H. Shapiro. Peristaltic pumping. Ann. Rev. Fluid. Mech. 3:13–36, 1971.CrossRefGoogle Scholar
  15. 15.
    Kaviany, M. Laminar flow through a porous channel bounded by isothermal parallel plates. Int. J. Heat Mass Transfer 28(4):851–858, 1985.CrossRefGoogle Scholar
  16. 16.
    Klingelhöfer, J., B. Conrad, R. Benecke, and B. Frank. Transcranial Doppler ultrasonography of carotid-basilar collateral circulation in subclavian steal. Stroke 19(8):1036–1042, 1988.CrossRefPubMedGoogle Scholar
  17. 17.
    Lennon, R., A. Byron, J. D. Humphries, M. J. Randles, A. Carisey, S. Murphy, D. Knight, P. E. Brenchley, R. Zent, and M. J. Humphries. Global analysis reveals the complexity of the human glomerular extracellular matrix. J. Am. Soc. Nephrol. 2014. doi: 10.1681/ASN.2013030233.Google Scholar
  18. 18.
    Liu, H., P. R. Patil, and U. Narusawa. On Darcy–Brinkman equation: Viscous flow between two parallel plates packed with regular square arrays of cylinders. Entropy 9:118–131, 2007.CrossRefGoogle Scholar
  19. 19.
    Masters, C. L., and K. Beyreuther. Amyloid-β production. In: Neurodegeneration: The Molecular Pathology of Dementia and Movement Disorders2nd, edited by D. W. Dickson, and R. O. Weller. Oxford: Wiley, 2011, pp. 92–96.CrossRefGoogle Scholar
  20. 20.
    Miosge, N. The ultrastructural composition of basement membranes in vivo. Histol. Histopathol. 16:1239–1248, 2001.PubMedGoogle Scholar
  21. 21.
    Mitchell, G. F., M. A. van Buchem, S. Sigurdsson, J. D. Gotal, M. K. Jonsdottir, O. Kjartansson, M. Garcia, T. Aspelund, T. B. Harris, V. Gudnason, and L. J. Launer. Arterial stiffness, pressure and flow pulsatility and brain structure and function: the Age, Gene/Environment Susceptibility—Reykjavik Study. Brain 134:3398–3407, 2011.CrossRefPubMedPubMedCentralGoogle Scholar
  22. 22.
    Preston, S. D., P. V. Steart, A. Wilkinson, J. A. R. Nicoll, and R. O. Weller. Capillary and arterial amyloid angiopathy in Alzheimer’s disease: Defining the perivascular route for the elimination of amyloid beta from the human brain. Neuropathol. Appl. Neurobiol. 29:106–117, 2003.CrossRefPubMedGoogle Scholar
  23. 23.
    Sangani, A. S., and A. Acrivos. Slow flow past periodic arrays of cylinders with application to heat transfer. Int. J. Multiphase Flow 8(3):193–206, 1982.CrossRefGoogle Scholar
  24. 24.
    Schley, D., R. Carare-Nnadi, C. P. Please, V. H. Perry, and R. O. Weller. Mechanisms to explain the reverse perivascular transport of solutes out of the brain. J. Theor. Biol. 238(4):962–974, 2006; (Epub 2005/08/23).CrossRefPubMedGoogle Scholar
  25. 25.
    Suleiman, H., L. Zhang, R. Roth, J. E. Heuser, H. Miner, A. S. Shaw, and A. Dani. Nanoscale protein architecture of the kidney glomerular basement membrane. eLife 2:e01149, 2013. doi: 10.7554/eLife.01149.PubMedPubMedCentralGoogle Scholar
  26. 26.
    Szentistvanyi, I., C. S. Patlak, R. A. Ellis, and H. F. Cserr. Drainage of interstitial fluid from different regions of rat brain 32. Am. J. Physiol. 246:6, 1984.Google Scholar
  27. 27.
    Viswanathan, A., and S. M. Greenberg. Cerebral amyloid angiopathy in the elderly. Ann. Neurol. 70(6):871–880, 2011; (Epub 2011/12/23).CrossRefPubMedPubMedCentralGoogle Scholar
  28. 28.
    Wang, P., and W. L. Olbricht. Fluid mechanics in the perivascular space. J. Theor. Biol. 274(1):52–57, 2011.CrossRefPubMedGoogle Scholar
  29. 29.
    Weller, R. O., D. Boche, and J. A. Nicoll. Microvasculature changes and cerebral amyloid angiopathy in Alzheimer’s disease and their potential impact on therapy. Acta Neuropathol. 118:87–102, 2009. doi: 10.1007/s00401-009-0498-z.CrossRefPubMedGoogle Scholar
  30. 30.
    Weller, R. O., E. Djuanda, H.-Y. Yow, and R. O. Carare. Lymphatic drainage of the brain and the pathophysiology of neurological disease. Acta Neuropathol. 117:1–14, 2009.CrossRefPubMedGoogle Scholar
  31. 31.
    Weller, R. O., C. A. Hawkes, R. O. Carare, and J. Hardy. Does the difference between PART and Alzheimer’s disease lie in the age-related changes in cerebral arteries that trigger the accumulation of Abeta and propagation of tau? Acta Neuropathol. 129:763–766, 2015. doi: 10.1007/s00401-015-1416-1.CrossRefPubMedGoogle Scholar
  32. 32.
    Weller, R. O., M. Subash, S. D. Preston, I. Mazanti, and R. O. Carare. Perivascular drainage of amyloid-beta peptides from the brain and its failure in cerebral amyloid angiopathy and Alzheimer’s disease. Brain Pathol. 18(2):253–266, 2008.CrossRefPubMedGoogle Scholar
  33. 33.
    Yamada, S., M. de Pasquale, C. S. Patlak, and H. F. Cserr. Albumin outflow into deep cervical lymph from different regions of rabbit brain. Am. J. Physiol. 261:H1197–H1204, 1991.PubMedGoogle Scholar

Copyright information

© Biomedical Engineering Society 2015

Authors and Affiliations

  • M. Keith Sharp
    • 1
  • 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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