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TECHNICAL PAPERS: Fluids/Heat/Transport

# One-Dimensional Models of the Human Biliary System

[+] Author and Article Information
W. G. Li, S. B. Chin

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

X. Y. Luo

Department of Mathematics, University of Glasgow, Glasgow, G12 8QW, UKX.Y.Luo@maths.gla.ac.uk

A. G. Johnson1

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

N. A. Hill

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

N. Bird

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

1

Deceased.

J Biomech Eng 129(2), 164-173 (Oct 07, 2006) (10 pages) doi:10.1115/1.2472379 History: Received February 01, 2005; Revised October 07, 2006

## Abstract

This paper studies two one-dimensional models to estimate the pressure drop in the normal human biliary system for Reynolds number up to 20. Excessive pressure drop during bile emptying and refilling may result in incomplete bile emptying, leading to stasis and subsequent formation of gallbladder stones. The models were developed following the group’s previous work on the cystic duct using numerical simulations. Using these models, the effects of the biliary system geometry, elastic property of the cystic duct, and bile viscosity on the pressure drop can be studied more efficiently than with full numerical approaches. It was found that the maximum pressure drop occurs during bile emptying immediately after a meal, and is greatly influenced by the viscosity of the bile and the geometric configuration of the cystic duct, i.e., patients with more viscous bile or with a cystic duct containing more baffles or a longer length, have the greatest pressure drop. It is found that the most significant parameter is the diameter of the cystic duct; a 1% decrease in the diameter increases the pressure drop by up to 4.3%. The effects of the baffle height ratio and number of baffles on the pressure drop are reflected in the fact that these effectively change the equivalent diameter and length of the cystic duct. The effect of the Young’s modulus on the pressure drop is important only if it is lower than $400Pa$; above this value, a rigid-walled model gives a good estimate of the pressure drop in the system for the parameters studied.

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## Figures

Figure 10

Variation of (a) the pressure drop ΔpCD and (b) the area ratio αout with flow rate for various values of Young’s modulus. All other parameters are chosen to be those in the Reference Set.

Figure 11

Variation of pressure drop ΔpCD with Young’s modulus. All other parameters are chosen to be those in the Reference Set.

Figure 12

Variation of the friction factor ratio with Reynolds number for cystic duct with rigid and elastic wall for (a) ξ=0.3 and 0.7 and (b) n=2 and 18. All other parameters are chosen to be those in the Reference set.

Figure 13

Variation of A1 and A2 with the number of baffles n.

Figure 1

Gross anatomy of the human biliary tree showing part of the gallbladder neck connected to the spiral valves in the cystic duct (3)

Figure 2

Schematic geometry model of human billiary system (a) and bile flow directions in the (b) emptying and (c) refill phases

Figure 3

Gallbladder volume variation with time during emptying and refilling. Note that only part of the refilling phase is plotted.

Figure 7

Pressure drop variation with time predicted using both rigid and elastic 1D models, all other parameters are chosen to be those in the Reference Set

Figure 8

Pressure drop variations with (a) baffle height ratio ξ, and (b) number of baffle n. All other parameters are chosen to be those in the Reference Set.

Figure 9

Pressure drop variations with (a) cystic duct diameter dCD and E=100Pa, and (b) bile viscosity ν and E=300Pa. All other parameters are chosen to be those in the Reference Set.

Figure 4

Baffle and cross sections of duct. A1 is the cross-sectional area of flow at point 1, and A2 the cross-sectional area of the flow at point 2.

Figure 5

A simplified cystic duct in the emptying phase. The duct is initially circular at the inlet, and the downstream part collapses due to the pressure drop as bile flows.

Figure 6

Comparison of the pressure drop estimated using the 1D rigid model (solid line) and the 3D numerical simulations (symbols). The 3D geometries of the cystic duct are taken from (25).

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