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Research Papers

# Mathematical Modeling of the Circulation in the Liver Lobule

[+] Author and Article Information
Andrea Bonfiglio

Department of Civil, Environmental and Architectural Engineering, University of Genoa, Via Montallegro 1, 16145 Genoa, Italyandrea.bonfiglio@unige.it

Department of Bioengineering, Imperial College London, London SW7 2AZ, UKk.leungchavaphongse08@imperial.ac.uk

Rodolfo Repetto

Department of Civil, Environmental and Architectural Engineering, University of Genoa, Via Montallegro 1, 16145 Genoa, Italyrodolfo.repetto@unige.it

Jennifer H. Siggers

Department of Bioengineering, Imperial College London, London SW7 2AZ, UKj.siggers@imperial.ac.uk

J Biomech Eng 132(11), 111011 (Oct 27, 2010) (10 pages) doi:10.1115/1.4002563 History: Received September 02, 2009; Revised August 25, 2010; Posted September 16, 2010; Published October 27, 2010; Online October 27, 2010

## Abstract

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.

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## Figures

Figure 5

Difference between the dimensionless pressure predicted by the shear-thinning model and the Newtonian model

Figure 6

(a) Solid displacement (arrows) and contour lines of the absolute value of the displacement when the portal pressure is increased by 15% from the physiological value, corresponding to δ=0.08. (b) Contour lines of K1∗/Kphys∗ in the deformed configuration. In both parts of the figure, we use ϵ=0.12.

Figure 7

Pressure-flux graph for the deformable model. Both axes have been scaled by their respective values in the physiological state. The straight solid thin line shows the behavior of the basic model. The empty circles represent the experimental results obtained by Ref. 5.

Figure 8

Analytical solution given in Eq. 2: (a) pressure field and (b) velocity field

Figure 1

Sketch of a cross-section of the domain showing the geometrical notation

Figure 2

(a) Pressure field, (b) velocity field (arrows), and contour lines of the velocity magnitude

Figure 3

Pressure (dashed curve) and velocity (solid curve) along two cuts through the domain: (a) along the line connecting the center of the portal tract [(x,y)=(−1,0)] to the center of the centrilobular vein [(x,y)=(0,0)] and (b) along a vascular septum (straight edge of the hexagon)

Figure 4

Difference between the dimensionless pressure in the anisotropic and isotropic cases: (a) kϕ/kr=1/2 and (b) kϕ/kr=1/3

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