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Hepatic MicrocirculationEnrico Bagnoli

Abstract

The liver is one of the organs most special and complex of the whole human body. Its functions arevital so the fully understanding of its operations is a crucial step in medical research. Previous studies haveclearly shown how the microcirculation significantly affects many hepatic processes such as liver rigenerationor drug absorption and clearence. However the complexity of the hemodynamics of this organ requires theuse of a model for this analysis. Aim of this work is the creation and the study of a simple and effectivemodel for the hepatic microcirculation.

I. INTRODUCTION

The Hepatic circulation is unique among vascularbeds. The most obvious unique features include: the dualvascular supply; the mechanism of intrinsic regulation ofthe hepatic artery (the hepatic arterial buffer response);the fact that portal blood flow, supplying two thirdsof liver blood flow, is not controlled directly by theliver; that 20% of the cardiac output rushes throughthe most vascularized organ in the body, driven by apressure gradient of only a few millimeters of mer-cury; the extremely distensible capacitance and venousresistance sites; the unidirectional acinar blood flow thatregulates parenchymal cell metabolic specialization; andthe high concentration of macrophagic (Kupffer) cellsfiltering the blood. The liver is the only organ reportedto have regional blood flow monitored by the autonomicnervous system. This mechanism, when dysfunctional,accounts for the hepatorenal syndrome and offers amechanistic therapeutic target to treat this syndrome. Thetrigger for liver regeneration is dependent on hepatichemodynamics so that chronic liver blood flow regulatesliver cell mass. In severe liver disease, the whole bodycirculation is reorganized, by forming portacaval shunts,to accommodate the increased intrahepatic venous resis-tance. These shunts protect the venous drainage of thesplanchnic organs but lead to loss of major regulatoryroles of the liver.

II. THE LIVER

The liver lies almost completely under the protectionof the rib-cage, projecting below it and coming intocontact with the anterior abdominal wall only belowthe right costal margin and the xiphisternum. The liverconsists of two main parts: a larger right lobe, a smallerleft lobe and two minor lobes. The upper border of theright lobe is at the level of the top of the 5th rib and theupper border of the left lobe is just below the 5th rib

Figure 1. Anatomical location of the liver in human body. Theadult human liver weights between 1.4 and 1.6 Kg. It measuresapproximately about 20 cm horizontally (across) and 17 cm verticallyand is 12 cm thick.

(Fig.1). It is a brownish-red organ and it is the largestinternal organ within the human body.

The microcirculation of the liver is of utmost im-portance for the physiology and function of the wholeorganism. It guarantees the supply of the parenchymaltissue with oxygen and nutrients, serves as the gatefor leukocyte entrance in hepatic inflammation, and isresponsible for the clearance of toxicants and foreignbodies from the bloodstream.

The liver is estimated to have over 500 functions. Themost important ones are summarised below.• Synthesise proteins, including albumin (to help

maintain the volume of blood) and blood clottingfactors.

• Synthesise, store, and process (metabolise) fats,including fatty acids (used for energy) and choles-terol.

• Metabolise and store carbohydrates, which are usedas the source for the sugar (glucose) in blood thatred blood cells and the brain use.

• Form and secrete bile that contains bile acids toaid in the intestinal absorption of fats and the fat-soluble vitamins A, D, E, and K.

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• Eliminate, by metabolising and/or secreting, thepotentially harmful biochemical products producedby the body.

The main vessels which constitute the vascular systemof the liver are:• the portal vein: drains blood from the digestive

system and its associated glands. Its main tributariesare the splenic vein and superior mesenteric vein. Itdivides into a right and a left branch before enteringthe liver distributing to the liver parenchyma thenutrients absorbed in the small intestine.

• the hepatic artery: is the blood vessel that suppliesoxygenated blood to the liver and it arises from theceliac trunk, a branch of the aorta.

• the hepatic veins: are the blood vessels that drainblood from the liver. They drain the blood into theinferior vena cava.

The liver receives 25% of the cardiac output, althoughit constitutes only 2.5% of body weight. The hepaticparenchymal cells are the most richly perfused of anyof the organs, and each parenchymal cell on the averageis in contact with perfusate on two sides of the cell. Ofthe total hepatic blood flow (100–130 ml/min per 100 gof liver, 30 ml/min per kilogram of body weight), onefifth to one third is supplied by the hepatic artery. Abouttwo thirds of the hepatic blood supply is portal venousblood. This is the most important difference between thehepatic circulatory system from that of other organs.

III. THE LOBULE

The liver lobes are made up of microscopic unitscalled lobules which are roughly hexagonal in shape andhave a diameter of about 1 mm. They constitute the basicmorphofunctional units of the liver (Fig.2). These lobulescomprise of rows of liver cells called hepatocytes (Fig.3)which radiate out from a central point. The hepatic cellsare in close contact with blood-filled sinusoids and alsolie adjacent to canaliculi into which bile is secreted. Theafferent branches ramify into vessels of ever decreasingcalibre. These branches of the portal vein and hepaticartery are present, together with bile ducts, within theso called portal tracts. At the centre of each lobule isa centrilobular vein while the portal tracts are at theangles of the hexagon. Blood flows from the vessels ofthe portal tracts towards the centrilobular veins within anetwork of converging tortuous sinusoids (Fig.4). Theseare thin walled, fenestrated capillaries situated betweenliver cell trabelulae. The portal vein brings blood richin nutrients from the gut, while the hepatic artery bringsoxygenated blood to the liver, which is especially neededby the cells of the bile ducts. This close relationship

Figure 2. The structure of the liver’s functional units or lobules.Blood enters the lobules through branches of the portal vein andhepatic artery proper, then flows through sinusoids.

Figure 3. Histological image of the lobular liver structure.

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Figure 4. Structure of the liver sinusoid. Note that the size of eachsinusoid is comparable to the size of red blood cells.

between the hepatocytes and surrounding blood enablesmany metabolic processes to take place. Blood flows outof the sinusoids into the central vein, removing detoxifiedsubstances and metabolic end products. The central veinultimately reunites with the hepatic vein transportingthese substances out of the liver. Bile that is producedby the hepatocytes drains into tiny canals called bilecanaliculi. These drain into bile ducts located aroundthe lobule perimeter. Bile drains from canaliculi throughcanals and ductules into progressively larger bile ducts.

IV. MODEL AND METHODS

Many processes in the liver are heavily influenced bythe hepatic circulation and microcirculation, among theseare:• Liver rigeneration: its trigger is strongly dependent

on the hepatic hemodynamics.• Small-for-size livers: can be created after either a

small-graft liver transplant or after a partial liverresection. The transplanted liver gradually grows tobecome normal size, but during the process damagecan occur due to the high rate of perfusion. Thisresults in hypertension in the hepatic portal vein.A shunt may be inserted between the portal andhepatic veins to allow some blood to bypass theliver. Knowledge of the relationship between thepressure drop across the liver and the blood flowis a prerequisite for a proper shunt design.

• Oxygen distribution and hepatocyte activity: thearrangement of several portal tracts around onecentral vein may have an important role to improvethe oxygen supply to hepatocytes. Hepatocytes be-have differently in different regions of the lobuleand a model of the blood perfusion of the liverlobule might help understanding the heterogeneityof hepatocyte activity.

• Drug absorption and clearance: one of the mainfunctions of the liver is to metabolise substances

in the blood. Understanding of the spatial drugconcentration after administration would help topredict clearance times and determine how muchdrug is absorbed in different regions of the liver.

To fully understand these mechanisms a deep under-standing of blood flow dynamics within the organ isneeded. Hence the need for a mathematical model forthe study of microcirculation within the lobule.

The model used is the one proposed by Bonfiglio etal. (2010). Flow in porous media as lobule’s sinusoidsis governed by the Laplace equation for the pressure:

O2p = 0 (1)

The Laplace equation is linear. This implies that theprinciple of superimposition of effects holds. In otherwords if many different solutions of the equations arefound, their linear combination is still a solution. Con-sidering a two-dimensional point source/sink of strengthQ, located in O. Q represents the volume flux per unitlength entering/exiting from the source/sink. If Q > 0the point is a source, if Q < 0 the point is a sink. Forsymmetry reasons the velocity is everywhere radial (withrespect to a coordinate system centred in O). Moreover,the flux through any circle I centred in O has to be equalto Q. Knowing that:

Q = 2πqrr (2)

where qr denotes the apparent velocity. Applying nowthe Darcy Law:

qr = −kµOp (3)

where µ is the dynamic viscosity of the fluid andk is the permeability of the medium ([k] = L2). Itquantifies how much “resistance” the solid provides tofluids flowing through it. The result is:

p = − Qµ

2πklog r + c (4)

where c is a constant.Assuming now that:

1) portal tracts are modelled as point sources;2) centrilobular veins are modelled as point sinks;

the system to be solved takes the following form:

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O2p =

−µQ+

k∆ hexagon angles (portal tracts)

−µQ−

k∆ hexagon centre (centrilobular vein)

0 everywhere else(5)

Op · n = 0 along the hexagon sides (6)

where Q+ and Q− denote the intensity of the sourcesand sinks, respectively and considering Q+ > 0 andQ− < 0. Moreover, in the above expressions ∆ is theDirac function and n denotes the unit vector normal tothe hexagon side.

Furthermore, considering that:

3) in a lattice of hexagons there are twice as manysources (angles of the hexagons) than sinks (cen-tres of the hexagons). In order for the mass to beconserved the intensity of a sink has to be twicethat of a source;

4) for the no-flux boundary conditions at the hexagonsides to be satisfied, an infinite number ofhexagons which tessellate the whole space can beadded;

5) the solution for the flow induced by a point sourceand sink is known;

The desired flow can be obtained summing up differentsolutions. Working in terms of dimensionless variablesand scaling as follows:

x∗ =xL

; q∗ =qU

; p∗ =p

µUL

k

where x denotes lengths, L is the length of the side ofthe lobule and U is the characteristic blood velocity inthe sinusoids. The above pressure scale is suggested bythe Darcy Law. Moreover, assuming that the flux per unitlength Q+ coming out of portal tracts can be written as:

Q+ = πUd (7)

where d is the diamater of a portal tract. In termsof the above dimensionless variables equation the Darcy

x*-4 -3 -2 -1 0 1 2 3 4 5

y*

-5

-4

-3

-2

-1

0

1

2

3

4

5

Figure 5. Hexagonal pattern created in Matlab. The red stars are thecenters and the blue crosses the vertices of the hexagons.

Law take the form:

p∗(x∗, y∗) =

limN→∞

[−

2N−∑i=1

d

2Llog(√

(x∗ − x∗+i )2 + (y∗ − y∗+i )2)

+

N−∑j=1

d

Llog(√

(x∗ − x∗−j )2 + (y∗ − y∗−j )2)]

(8)

q∗ = −O∗p∗ (9)

where N− denotes the number of sinks (centres of thehexagons) and N+ = 2N− the number of sources (ver-tices of the hexagons), (x∗+i , y∗+i ) denotes the positionof the i-th source and, (x∗−j , y∗−j ) the position of the j-thsink.

V. RESULTS

The model has been implemented in Matlab (TheMathWorks, Inc., Natick, Massachusetts, United States.).Dimensional values used are the following:• Length of the side of a lobule L: 500 µm• Diameter of portal tracts d: 50 µm• Characteristic velocity in the sinusoidal space U

(for the rat): 4 · 10−3m/s

First of all an hexagonal grid has been created tosimulate the hepatic structure and the lobule morphology(Fig.5).Later the pression field (Fig.6) and the velocity field(Fig.7) have been calculated using the dimensionlessDarcy Law. The results clearly show the flux from theportal venules situated at the vertices of the hexagons

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x*-1 -0.5 0 0.5 1

y*

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 6. Pressure gradient function of x∗ and y∗ within the lobule.Cool colors represent weak pressure conversely warm colors representgreatest pressure.

x*-1 -0.5 0 0.5 1

y*

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 7. Velocity field function of x∗ and y∗ within the lobule.

to the central vein in the middle of the lobule. The fluxis contolled by a pressure gradient. Despite in absoluteterms the pressure field is quite low with a value of afew millimeters of mercury, working with dimensionlessvariables the different pressures of portal and central veinis very clear. In the vertices of the lobules pressure ismuch higher than in the center as shown in Fig.8.

Finally both pressure and velocity field are presentedin Fig.9 for more than one lobule, showing a goodprecision of the model for more hexagonal cells.

VI. CONCLUSION

In conclusion it can be said that the model and theapproximations used are acceptable. The model proposed

10.5

x*

0-0.5

-1-1

-0.5

0

y*

0.5

8.8

8.9

9

9.1

9.4

9.3

9.2

1

pres

sion

p*

Figure 8. The dimensionless function p∗(x∗, y∗) displays the trendof pressure within the lobule.

x*-1.5 -1 -0.5 0 0.5 1 1.5

y*

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Figure 9. Both the pressure gradient and the velocity field for morethan one lobule plotted together.

by Bonfiglio et al. despite its simplicity is suitable forstudies of this type. The results obtained are consistentand in line with the expected. The work approach usingDarcy Law and the geometric patterning of the livertissue has not proved limiting in the study of microcir-culation. Note that considering the blood rheology, onecan give a slightly more accurate distribution, especiallyconsidering the shear-thinning behavior of blood. Thismodification is useful in the implementation of a smallliver and the shunt design.

Eventually work with dimensionless and relative termsturns out to be advantageous both from the numericalpoint of view, that of clarity and readability of the result.

Future studies will be of interest to the link betweenregeneration, microcirculation and liver perfusion of vital

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importance in liver transplants.

REFERENCES

[1] The liver circulation and anatomy.[2] BONFIGLIO, A., LEUNGCHAVAPHONGSE, K.,

REPETTO, R., AND SIGGERS, J. H. Mathematicalmodeling of the circulation in the liver lobule.Journal of biomechanical engineering 132, 11(2010), 111011.

[3] JACOB, B. Dynamics of fluids in porous media. SoilScience (1972).

[4] LAUTT, W. W. Blood flow regulation of hepatocyteproliferation.

[5] LAUTT, W. W. Hepatic circulation: physiology andpathophysiology. In Colloquium Series on Inte-grated Systems Physiology: from Molecule to Func-tion (2009), vol. 1, Morgan & Claypool Publishers,pp. 1–174.

[6] LAUTT, W. W. Integrative hepatic response tohemorrhage.

[7] LEE, J., AND SMITH, N. P. Theoretical modeling inhemodynamics of microcirculation. Microcirculation15, 8 (2008), 699–714.

[8] REPETTO, R. Notes on Biofluid Dynamics, 2014.