In this paper, we develop a mathematical model of blood circulation in the liver lobule. We aim to find the pressure and flux distributions within a liver lobule. We also investigate the effects of changes in pressure that occur following a resection of part of the liver, which often leads to high pressure in the portal vein. The liver can be divided into functional units called lobules. Each lobule has a hexagonal cross-section, and we assume that its longitudinal extent is large compared with its width. We consider an infinite lattice of identical lobules and study the two-dimensional flow in the hexagonal cross-sections. We model the sinusoidal space as a porous medium, with blood entering from the portal tracts (located at each of the vertices of the cross-section of the lobule) and exiting via the centrilobular vein (located in the center of the cross-section). We first develop and solve an idealized mathematical model, treating the porous medium as rigid and isotropic and blood as a Newtonian fluid. The pressure drop across the lobule and the flux of blood through the lobule are proportional to one another. In spite of its simplicity, the model gives insight into the real pressure and velocity distribution in the lobule. We then consider three modifications of the model that are designed to make it more realistic. In the first modification, we account for the fact that the sinusoids tend to be preferentially aligned in the direction of the centrilobular vein by considering an anisotropic porous medium. In the second, we account more accurately for the true behavior of the blood by using a shear-thinning model. We show that both these modifications have a small quantitative effect on the behavior but no qualitative effect. The motivation for the final modification is to understand what happens either after a partial resection of the liver or after an implantation of a liver of small size. In these cases, the pressure is observed to rise significantly, which could cause deformation of the tissue. We show that including the effects of tissue compliance in the model means that the total blood flow increases more than linearly as the pressure rises.

1.
Garcea
,
G.
, and
Maddern
,
G. J.
, 2009, “
Liver Failure After Major Hepatic Resection
,”
J. Hepatobiliary Pancreat Surg.
0944-1166,
16
(
2
), pp.
145
155
.
2.
Kiernan
,
F.
, 1833, “
The Anatomy and Physiology of the Liver
,”
Philos. Trans. R. Soc. London
0962-8428,
123
, pp.
711
770
.
3.
Teutsch
,
H. F.
,
Schuerfeld
,
D.
, and
Groezinger
,
E.
, 1999, “
Three-Dimensional Reconstruction of Parenchymal Units in the Liver of the Rat
,”
Hepatology (Philadelphia, PA, U. S.)
0270-9139,
29
(
2
), pp.
494
505
.
4.
Teutsch
,
H.
, 2005, “
The Modular Microarchitecture of Human Liver
,”
Hepatology (Philadelphia, PA, U. S.)
0270-9139,
42
(
2
), pp.
317
325
.
5.
Dahmen
,
U.
,
Hall
,
C. A.
,
Madrahimov
,
N.
,
Milekhin
,
V.
, and
Dirsch
,
O.
, 2007, “
Regulation of Hepatic Microcirculation in Stepwise Liver Resection
,”
Acta Gastroenterol. Belg.
0001-5644,
70
(
4
), pp.
345
51
.
6.
Rani
,
H.
,
Sheu
,
T. W.
,
Chang
,
T.
, and
Liang
,
P.
, 2006, “
Numerical Investigation of Non-Newtonian Microcirculatory Blood Flow in Hepatic Lobule
,”
J. Biomech.
0021-9290,
39
(
3
), pp.
551
563
.
7.
Van Der Plaats
,
A.
,
Hart
,
N. A.
,
Verkerke
,
G. J.
,
Leuvenink
,
H. G. D.
,
Verdonck
,
P.
,
Ploeg
,
R. J.
, and
Rakhorst
,
G.
, 2004, “
Numerical Simulation of the Hepatic Circulation
,”
Int. J. Artif. Organs
0391-3988,
27
(
3
), pp.
222
230
.
8.
Ricken
,
T.
,
Dahmen
,
U.
, and
Dirsch
,
O.
, 2010, “
A Biphasic Model for Sinusoidal Liver Perfusion Remodeling After Outflow Obstruction
,”
Biomech. Model. Mechanobiol.
1617-7959,
9
(
4
), pp.
435
450
.
9.
Popel
,
A. S.
, and
Johnson
,
P. C.
, 2005, “
Microcirculation and Hemorheology
,”
Annu. Rev. Fluid Mech.
0066-4189,
37
, pp.
43
69
.
10.
Lee
,
J.
, and
Smith
,
N. P.
, 2008, “
Theoretical Modeling in Hemodynamics of Microcirculation
,”
Microcirculation (Philadelphia)
1073-9688,
15
(
8
), pp.
699
714
.
11.
Huyghe
,
J. M.
, and
van Campen
,
D. H.
, 1995, “
Finite Deformation Theory of Hierarchically Arranged Porous Solids—I. Balance of Mass and Momentum
,”
Int. J. Eng. Sci.
0020-7225,
33
(
13
), pp.
1861
1871
.
12.
Huyghe
,
J. M.
, and
van Campen
,
D. H.
, 1995, “
Finite Deformation Theory of Hierarchically Arranged Porous Solids—II. Constitutive Behaviour
,”
Int. J. Eng. Sci.
0020-7225,
33
(
13
), pp.
1873
1886
.
13.
Vankan
,
W. J.
,
Huyghe
,
J. M.
,
Janssen
,
J. D.
, and
Huson
,
A.
, 1996, “
Poroelasticity of Saturated Solids With an Application to Blood Perfusion
,”
Int. J. Eng. Sci.
0020-7225,
34
(
9
), pp.
1019
1031
.
14.
Schmid-Schönbein
,
H.
, 1988, “
Conceptual Proposition for a Specific Microcirculatory Problem: Maternal Blood Flow in Hemochorial Multivillous Placentae as Percolation of a Porous Medium
,”
Trophoblast Research
,
P.
Kaufman
and
R. K.
Miller
, eds.,
Plenum
,
New York
, Vol.
3
, pp.
17
38
.
15.
Chernyavsky
,
I. L.
,
Jensen
,
O. E.
, and
Leach
,
L.
, 2010, “
A Mathematical Model of the Intervillous Blood Flow in the Human Placentone
,”
Placenta
0143-4004,
31
(
1
), pp.
44
52
.
16.
Wangyi
,
W.
, and
Xiaoyi
,
H.
, 1988, “
A Continuum Model of the Blood Flow in the Lung Microcirculation
,”
Acta Mech. Sin.
0459-1879,
4
(
3
), pp.
230
238
.
17.
Sivaloganathan
,
S.
,
Stastna
,
M.
,
Tenti
,
G.
, and
Drake
,
J.
, 2005, “
Biomechanics of the Brain: A Theoretical and Numerical Study of Biot’s Equations of Consolidation Theory With Deformation-Dependent Permeability
,”
Int. J. Non-Linear Mech.
0020-7462,
40
(
9
), pp.
1149
1159
.
18.
Simon
,
B. R.
,
Liable
,
J. P.
,
Pflaster
,
D.
,
Yuan
,
Y.
, and
Krag
,
M. H.
, 1996, “
A Poroelastic Finite Element Formulation Including Transport and Swelling in Soft Tissue Structures
,”
ASME J. Biomech. Eng.
0148-0731,
118
(
1
), pp.
1
9
.
19.
Lipowsky
,
H. H.
, and
Zweifach
,
B. W.
, 1974, “
Network Analysis of Microcirculation of Cat Mesentery
,”
Microvasc. Res.
0026-2862,
7
(
1
), pp.
73
83
.
20.
Pries
,
A. R.
,
Secomb
,
T. W.
,
Gaehtgens
,
P.
, and
Gross
,
J. F.
, 1990, “
Blood Flow in Microvascular Networks. Experiments and Simulation
,”
Circ. Res.
0009-7330,
67
(
4
), pp.
826
834
.
21.
Bear
,
J.
, 1988,
Dynamics of Fluids in Porous Media
,
Dover
,
New York
.
22.
Smye
,
S. W.
,
Evans
,
C. J.
,
Robinson
,
M. P.
, and
Sleeman
,
B. D.
, 2007, “
Modelling the Electrical Properties of Tissue as a Porous Medium
,”
Phys. Med. Biol.
0031-9155,
52
(
23
), pp.
7007
7022
.
23.
Burt
,
A. D.
,
Portmann
,
B. C.
, and
Ferrell
,
L. D.
, 2006,
MacSween’s Pathology of the Liver
,
5th ed.
,
Churchill Livingstone, Elsevier
,
Philadelphia, PA
.
24.
Shibayama
,
Y.
, and
Nakata
,
K.
, 1985, “
Localization of Increased Hepatic Vascular Resistance in Liver Cirrhosis
,”
Hepatology (Philadelphia, PA, U. S.)
0270-9139,
5
(
4
), pp.
643
648
.
25.
Fauci
,
A. S.
,
Braunwald
,
E.
,
Kasper
,
D. L.
,
Hauser
,
S. L.
,
Longo
,
D. L.
,
Jameson
,
J. L.
, and
Loscalzo
,
J.
, 2008,
Harrison’s Principles of Internal Medicine
,
17th ed.
,
McGraw-Hill
,
New York
.
26.
Dal Santo
,
G.
, 1993,
A Laboratory Basis for Anesthesiology
,
Piccin
,
Padova, Italy
.
27.
Koo
,
A.
,
Liang
,
I. Y. S.
, and
Cheng
,
K. K.
, 1975, “
The Terminal Hepatic Microcirculation in the Rat
,”
Q. J. Exp. Physiol. Cogn Med. Sci.
0033-5541,
60
, pp.
261
266
.
28.
Chui
,
C.
,
Kobayashi
,
E.
,
Chen
,
X.
,
Hisada
,
T.
, and
Sakuma
,
I.
, 2004, “
Combined Compression and Elongation Experiments and Non-Linear Modelling of Liver Tissue for Surgical Simulation
,”
Med. Biol. Eng. Comput.
0140-0118,
42
(
6
), pp.
787
798
.
29.
Walburn
,
F. J.
, and
Schneck
,
D. J.
, 1976, “
A Constitutive Equation for Whole Human Blood
,”
Biorheology
0006-355X,
13
(
3
), pp.
201
210
.
30.
Corapcioglu
,
M.
, 1996,
Advances in Porous Media
,
1st ed.
,
Elsevier
,
New York
, Vol.
3
.
31.
Chui
,
C.
,
Kobayashi
,
E.
,
Chen
,
X.
,
Hisada
,
T.
, and
Sakuma
,
I.
, 2007, “
Transversely Isotropic Properties of Porcine Liver Tissue: Experiments and Constitutive Modelling
,”
Med. Biol. Eng. Comput.
0140-0118,
45
, pp.
99
106
.
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