visualization experiment of flow structures inside two-dimensional human biliary system models

October 4, 2006 17:31 WSPC/170-JMMB 00191

Journal of Mechanics in Medicine and BiologyVol. 6, No. 3 (2006) 249–260c© World Scientific Publishing Company

VISUALIZATION EXPERIMENT OF FLOW STRUCTURESINSIDE TWO-DIMENSIONAL HUMAN BILIARY SYSTEM

MODELS

M. AL-ATABI∗,†, S. B. CHIN† and X. Y. LUO‡†Department of Mechanical Engineering,

University of Sheffield, Sheffield S1 3JD, South Yorkshire, UK‡Department of Mathematics, University of Glasgow, Glasgow, UK

∗[email protected]

Received 4 October 2005Accepted 28 November 2005

Although little is known about the transport mechanism of bile in the human biliary sys-tem, clinical studies suggest that it may be a contributing factor in the pathogenesis ofcholelithiasis. This paper reports an investigation of the steady flow in two-dimensionalhuman biliary system models using flow visualization. The geometries of these two-dimensional models were described based on actual patients’ operative cholangiograms.Two models were used: one represented biliary system of a patient suffering from gall-stones, while the second represented the biliary system of a healthy person. The stream-lines of the flow in the cystic duct of the patient suffering from gallstones were highlywinding, compared with those of the healthy person. This is an indication of a higherresistance to the flow in the cystic duct of the gallstone patient, which is a contributingfactor of the formation of gallstones. The work presented here is a part of an ongoingproject aimed at understanding the functions of human cystic duct and the role of bileflow in the formation of gallstones.

Keywords: Gallstones; bile flow; cystic duct; flow visualization.

Nomenclature

Dh hydraulic diameter (m)H height of the channel (m)∆P pressure drop (Pa)Q volumetric flow rate (m3 s−1)Re Reynolds numberR = ∆P/Q cystic duct resistance (kgm−4 s−1)V average velocity (m s−1)W width of the channel (m)µ dynamic viscosity (Pa s)ρ density (kg m−3)

∗Corresponding author.

249


250 M. Al-Atabi, S. B. Chin & X. Y. Luo

1. Introduction

The human biliary system is responsible for creating, transporting, storing andreleasing bile into the duodenum to aid in the digestion of fats. The biliary anatomycomprises the liver, gallbladder and biliary tract (cystic duct, hepatic duct, andcommon bile duct) as depicted in Fig. 1. Bile is an aqueous fluid secreted by theliver into the biliary tracts. Similar to most biological fluids, it is a complex mixtureof biochemicals, bio-fluids and solids, and is modified by secretion and/or absorp-tion during its passage to the gallbladder and during storage in the gallbladder.The gallbladder is unique among other hollow human organs in only having a singleconduit (cystic duct) for filling and emptying. Gallstone formation (cholelithiasis)and gallbladder inflammation (cholecystitis) are the most common biliary disor-ders. Although these conditions are rarely life threatening, they cause considerabledeterioration in the quality of life in those so afflicted, and operations to remove dis-eased gallbladder (cholecystectomy) are the most commonly performed abdominaloperations in developed countries.1

The three main factors in the pathophysiological genesis of cholelithiasis arebelieved to be:

(1) super-saturation of bile with cholesterol,(2) presence of calculi nucleating agents, and(3) reduction in gallbladder motility.

The first two factors have been studied extensively, but Holzbach et al.2 reportedthat supersaturated bile is frequently encountered in healthy individuals. Hence,

Fig. 1. Anatomical details of a human biliary system.


Experiment of Flow Structures Inside 2D Human Biliary System Models 251

recent studies have focused on the role of bile transport in the biliary system andmany clinical studies support the hypothesis that the fluid mechanics of the biliarysystem might be an important contributory factor in the formation of gallstones.Amongst them, Deenitchin et al.3 statistically showed that cholelithiasis is linkedto abnormal cystic duct configuration and those with gallstones had longer or nar-rower cystic ducts than those without them; Holzbach et al.2 and Catnach et al.4

found that prolonged stasis of bile in the gallbladder contributed significantly togallstone formation; and patients with cystic duct syndrome are found to have alow gallbladder ejection fraction.5

A comprehensive study to ascertain the pathophysiological functions of the fluidmechanics of the biliary system is needed. Ooi et al.6–8 employed CFD model-ing to determine the role of cystic duct acting as a sphincter. Bile flow throughtwo- and three-dimensional models of idealized and actual cystic ducts has beensimulated to characterize the influence of anatomical structure on flow resistance.Al-Atabi et al.9–12 used experimental pressure drop measurement and flow visu-alization to investigate the flow of bile in idealized cystic duct models made oftransparent circular tubes with segmental baffles. The numerical simulations andthe experimental investigation demonstrated the relationship between flow struc-ture and the anatomy of the cystic duct, and the corresponding resistance to flowfrom the gallbladder. The greater the complexity of the cystic duct geometry, thegreater is the pressure drop and the resistance to flow.

The present paper reports on the initial stages of experimental programme thataims to develop non-invasive techniques to provide insight into the role of fluidmechanics in the pathogenesis of gallstone formation and the origins of biliarypains, and to provide additional diagnostic tools to physicians. Two-dimensionalmodels of anatomically faithful rigid cystic ducts have been constructed. Flow ratemeasurements and flow visualization have been made.

2. Materials and Methods

2.1. Cystic duct models

Figure 2 shows the complexity of the anatomy of the human cystic duct, whichis connected to the gallbladder by a neck. It can be clearly seen that the ductcomprises the “valves of Heister,” which are a series of semi-lunar or crescent-shaped folds. The number of such folds, the length, and the diameter of the ductvaries between individuals, typically between 2 and 14, 10 and 60mm, and 2 and5mm, respectively. There is also variation in the curvature of the cystic duct. Athin layer of smooth muscle covers the length of the duct.

Two cystic duct models were studied for comparison: one belonging to a patientsuffering from gallstones (Patient A) (Fig. 3(a)) and the other belonging to a healthyperson (Patient B) (Fig. 3(b)). These models were identical to the models numeri-cally studied by Ooi et al.8



(a) (b)

Fig. 2. Operative cholangiograms of (a) Patient A with gallstones and (b) Patient B withoutgallstones (images kindly provided by Department of Radiology, Royal Hallamshire Hospital,Sheffield). (Magnification factor ×1.)

The outline of enlarged (5×) two-dimensional image of each biliary system wasdrawn on a 50mm thick, non-porous Styrofoam board (Roofmate c© produced byDow Chemical Company). The profile was then cut using a hot wire to create theflow passages. An inlet channel was introduced to the fundus side of the gallbladderto create an emptying effect. Holes were made on the sides of the board to con-nect the gallbladder and the bile duct channels to the water supply, allowing thesimulation of both the filling and emptying of the gallbladder.

Transparent acrylic sheets of 8 mm thickness were used to cover both sides ofthe Styrofoam board. Layers of silicone glue were applied between the acrylic sheetsand the Styrofoam board. To secure the system and to prevent any leakage, thethree layers were bolted together with M8 bolts and nuts. Two models were createdto represent the biliary system of a healthy person (Patient B) and that of a personwith gallstones (Patient A). This is shown in Figs. 3 and 4, respectively.

To ensure that the flow in the cystic duct model is as near to two-dimensionalflow as possible, an average aspect ratio (AR = W/H) of 5 was chosen for thecross-section of the flow passage.13 For two-dimensional channel flows, the Reynoldsnumber is calculated based on hydraulic diameter as follows

ReDh =ρV Dh

µ, (1)



Valve A

Valve B

Valve C

630 mm

580 mm

View A

Acrylic Board

Styrofoam

Acrylic Board

M 8 BoltMetal Washer

M 8 Nut

View A

H

Fig. 3. Two-dimensional rigid model for biliary system of a healthy person.

370 mm

610 mm

Valve A

Valve B

Valve C

Fig. 4. Two-dimensional rigid model for biliary system of a person suffering from gallstones.

where

Dh = limH→∞

4HW

2 (H + W )= lim

(W/H)→0

2W

1 + (W/H)= 2W. (2)

2.2. Rheological properties of bile

A literature search reveals that there are few convincing rheological data on humanbile. Gottschalk and Lochner14 reported from 33 samples that post-operative



hepatic human bile is a Maxwell fluid. Coene et al.15 found that nearly a thirdof their hepatic bile samples displayed non-Newtonian behavior. Rodkiewicz andOtto16 reported that bovine bile flowing through a rigid tube behaved like aNewtonian fluid, but Rodkiewicz et al.,17 by relating measured pressure drop toflow rate, also showed that the flow of bile in a duct of the extrahepatic biliary treeis non-Newtonian.

The present research group carries out an ongoing programme to measure therheological properties of human bile. Preliminary measurements of fresh bile suggestthat for a healthy person without gallstones, the gallbladder and hepatic bile is aNewtonian fluid with a constant viscosity of ∼1mPa s and a density of nearly1000 kgm−3.18 Since these values are close to those for water, filtered city waterwas used as the test fluid in this study.

2.3. Flow assumptions

Human gallbladder empties at an average rate of 1ml min−1, as suggested byultrasonographic imaging of the rate of change of the gallbladder volume.19 Theaverage flow rate in the duct is ∼0.5–1.0ml min−1 for fasting gallbladders and2.0–3.0ml min−1 after meal.20 The Reynolds number, Re, based on mean velocityand mean duct diameter, thus varies from 1 to 40. In addition, as the average timeto empty a gallbladder with a mean volume of 35ml of bile is ∼30min,21 the exper-iments assume that the flow is laminar and that it is sufficiently slow to considerthat steady-state conditions prevail.

2.4. Flow circuit

The biliary system models were used in a flow circuit shown diagrammatically inFig. 4. Steady flow was provided by a 40-l constant-head tank. Water was suppliedto the tank from the laboratory pipeline through a float. The inlet section intothe model was a straight PVC pipe of diameter 25.4mm, which incorporated a dieinjection needle at its geometric center. The flow was collected at the outlet tubeand the actual mean flow rates were determined by measuring the volume of fluidcollected over a known time interval.

The filling of the gallbladder was simulated by closing valves B and C, andusing valve A as an inlet and valve D as an outlet. Emptying of the gallbladder wassimulated by closing valves A and D, and using valve B as an inlet and valve C asan outlet as shown in Fig. 4.

2.5. Flow visualization

Flow visualization was realized by a fluorescent yellow/green dye, Fluorescen (pro-duced by Cole-Parmer instrument company).22 The dye is neutrally buoyant in



Constant head tank

Valve A

Valve B

Valve C

Settling Length

Reservoir

Watersupply

Valve D

Fig. 5. Schematic diagram of experimental set-up for flow visualization.

water. It was allowed to flow under gravity from a burette through a hypodermicneedle located upstream of the inlet of the test section. The outlet of the hypoder-mic needle was positioned at the geometric center of the pipe. A Sony DSC-S75digital camera with a 3.2 megapixel resolution was employed to record the flowstructures.

3. Results and Discussion

Due to the rigid wall and steady flow assumptions, the present models do notsimulate the contractile activity of gallbladder. The flow simulated here is similarto that of cholangiogram contrast medium, which is usually injected through thefundus. Figure 2 shows that Patient A had a longer but more uniform cystic duct,with more folds than those in Patient B. The latter duct also narrows considerablybetween the gallbladder neck and the hepatic duct, making it more like a nozzle, astructure known to have small flow losses and hence resistance.

The dimensionless flow resistance of these two patients’ biliary systems werenumerically computed by Ooi et al.8 over a range of Reynolds number as shownin Fig. 6. It is obvious that the resistance of Patient A is higher than that ofPatient B for all values of Re computed. Therefore, in order to maintain the sameflow, Patient A would have required a greater pressure difference between the gall-bladder and the common bile duct than that for Patient B. Since Patient A had



Fig. 6. Relationship between the Reynolds number and the flow resistance in the cystic ducts ofPatients A (with gallstones) and B (without gallstones).8

gallstones but not Patient B, these results suggest a tentative relationship betweenbile flow resistance, anatomical structure of the cystic duct, and the presence ofgallstones.

The flow visualization was performed after steady-state conditions prevailed atmass flow rates corresponding to Reynolds numbers of 20, 30, 40, and 60 (based onthe average duct hydraulic diameter) for both the gallbladder emptying and fillingconditions.

The results of this study are summarized in Figs. 7 (for Patient A) and 8(for Patient B). The left-hand columns in these figures represent the gallblad-der emptying, while the right-hand columns represent the gallbladder filling. Itis obvious that the streamlines in Fig. 7 are winding (due to the complicatedgeometry); this is an indication of a higher resistance to the flow, which is a pre-cursor for the formation of gallstones. On the other hand, in Fig. 8, the stream-lines of the flow appear to be rather straight, indicating a small resistance to theflow.

4. Conclusions

Flow visualization study has been made with rigid two-dimensional cystic ductmodels. The experiments form part of an overall programme to investigate the roleof fluid mechanics in gallstone formation. This study gives a qualitative indicationof the higher flow resistance in cystic ducts of patients with gallstones and agreeswith the recently published data of Ooi et al.8 Increased cystic duct resistance may



Re = 20 Re = 20

Re = 30 Re = 30

Re = 40 Re = 40

Re = 60 Re =60

Fig. 7. Flow visualization in 2D rigid biliary system of person with gallstones. Left column (gall-bladder emptying) and right column (gallbladder filling).



Re = 40 Re = 40

Re = 60 Re =60

Re = 20 Re = 20

Re = 30 Re = 30

Fig. 8. Flow visualization in 2D rigid biliary systems of healthy person. Left column (gallbladderemptying) and right column (gallbladder filling).

cause prolonged stasis of bile in the gallbladder, which may promote the formationof gallstone. Also, further work is required to obtain realistic 3D cystic duct andbiliary systems model, the effects of compliant boundaries, and non-Newtonianbehavior of gallbladder and hepatic bile.



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visualization experiment of flow structures inside two-dimensional human biliary system models

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