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Non-Newtonian Bile Flow in Elastic Cystic Duct: One- and Three-Dimensional Modeling

Annals of Biomedical Engineering volume 36, Article number: 1893 (2008) Cite this article

Abstract

Bile flow is thought to play an essential role in the pathophysiological genesis of cholelithiasis (gallstone formation) and in gallbladder pain. In this paper, we extend our previous study of the human biliary system (Li et al., 2007, J. Biomech. Eng., 129:164–173) to include two important factors: the non-Newtonian properties of bile, and elastic deformation of the cystic duct. A one-dimensional (1D) model is analyzed and compared with three-dimensional (3D) fluid–structure interaction simulations. It is found that non-Newtonian bile raises resistance to the flow of bile, which can be augmented significantly by the elastic deformation (collapse) of the cystic duct. We also show that the 1D model predicts the pressure drop of the cystic duct flow well for all cases considered (Newtonian or non-Newtonian flow, rigid or elastic ducts), when compared with the full 3D simulations.

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Notes

  1. ADINA’s second-order upwind scheme can be achieved by using the FCBI-C element, which, however, converges slowly for the deformed domain problems.

  2. Calculated based on σ = ΔPd/2h, where σ is the circumferential normal stress and ΔP is the pressure drop.

Abbreviations

A :

Cross-sectional area of collapsed duct, m2

A eq :

Cross-sectional area of duct at zero transmural pressure \( A_{{{\text{eq}}}} = {\pi d^{2}_{{{\text{eq}}}} } /4\), m2

A 1 :

Cross-sectional area of baffle clearance, m2

c 2 :

Sudden expansion head-loss coefficient

c 4 :

Head-loss coefficient for a 90° bend

d :

Internal diameter of duct, mm

E :

Young’s modulus of cystic duct wall, Pa

f :

Darcy friction factor

f 0 :

Darcy friction factor for duct without baffles

h :

Wall or baffle thickness, mm

H :

Baffle height, mm

k, l :

Exponent in tube laws

L :

Length of duct, m

L k :

Extra length due to minor pressure losses, m

m :

Power in the Carreau model

n :

Number of baffles or exponent in tube law

n c :

Maximum number of baffles considered

p :

Internal duct pressure, Pa

p e :

External duct pressure, Pa

Q :

Bile flow rate, mL/min

Re:

Reynolds number based on d eq and μ ∞

u :

Bile velocity in cystic duct, u = Q/A, m/s

V :

Gallbladder volume, mL

x, y, z :

Cartesian coordinates, m

α :

Area ratio, α = A/A eq

γ :

Shear rate, s−1

λ :

Time constant

ε :

Velocity deformation rate or shear strain rate, s−1

μ :

Bile dynamic viscosity, Pa s

μ 0 :

Dynamic viscosity at zero shear rate, Pa s

μ ∞ :

Dynamic viscosity at infinite shear rate, Pa s

κ :

Poisson’s ratio of the cystic duct wall

ρ :

Density of bile, kg/m3

τ :

Shear stress, Pa

ξ :

Dimensionless baffle height, ξ = H/d CD

Δp :

Pressure drop, Pa

Δp te :

Minor pressure drop in T-junction during emptying, Pa

Δp k :

Pressure losses due to bends, sudden expansion and contraction, Pa

b:

Baffle

CBD:

Common bile duct

CD:

Cystic duct

CHD:

Common hepatic duct

EM:

Emptying

eq:

Equivalent

i :

Coordinate index, i = 1, 2, 3

in:

Inlet of duct

j :

Coordinate index, j = 1, 2, 3

min:

Minimum value

out:

Outlet

RF:

Refilling

w:

On wall

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Acknowledgment

XYL would like to acknowledge a Global Research Award from the UK Royal Academy of Engineering.

Author information

Affiliations

  1. Department of Mathematics, University of Glasgow, Glasgow, G12 8QW, UK

    W.G. Li, X.Y. Luo & N.A. Hill

  2. Department of Mechanical Engineering, University of Sheffield, Sheffield, S1 3JD, UK

    S.B. Chin

  3. Academic Surgical Unit, Royal Hallamshire Hospital, Sheffield, S10 2JF, UK

    A.G. Johnson & N.C. Bird

Authors
  1. W.G. Li
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  2. X.Y. Luo
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  3. S.B. Chin
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  4. N.A. Hill
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Correspondence to X.Y. Luo.

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Li, W., Luo, X., Chin, S. et al. Non-Newtonian Bile Flow in Elastic Cystic Duct: One- and Three-Dimensional Modeling. Ann Biomed Eng 36, 1893 (2008). https://doi.org/10.1007/s10439-008-9563-3

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  • DOI: https://doi.org/10.1007/s10439-008-9563-3

Keywords

  • Gallbladder
  • Gallstone
  • Cystic duct
  • Biliary system
  • Non-Newtonian fluid
  • Pressure drop
  • Fluid–structure interaction
  • 1D modeling
  • Finite element methods