UK Nonlinear News , August 1998

A Computer-assisted Model to Visualise Liver Parenchyma Development

By N. Dioguardi1, A. Barbieri2, F.Grizzi1 & R. Dacc3

1Istituto Clinico Humanitas, Rozzano, Milano, Italy,

2 Centro Ricerche M.Branca, Milano, Italy,

3Ospedale S.Giuseppe, Milano, Italy


The liver has been described as a fractal structure made up of many self-similar elements called hepatones [1, 2]. The hepatone has been defined as the smallest part of whole liver action included in the smallest possible space, and is generated by the interrelationship between the hepatocyte and the three mesenchymal cells of liver tissue: the macrophagic Kupffer cell, the cell of the sinusoidal wall and the Ito cell [1, 2].

The hepatic vascular framework, as are many other systems such as the body vascular system, the lung airway or the neuronal shape network, is an example of natural fractal object. Every fractional branch of the hepatic vascular system is characterised by irregular forms resembling the structural complexity of the whole, whatever the scale used for the observation. This tree-like morphology is repeated from the trunk of the portal and hepatic veins and hepatic artery until the finest terminal branches.

The liver parenchyma is made up of interlacing plates of adjacent hepatocytes covered by a thin layer of endothelial lining cells that demarcate blood sinusoids.

The normal morphological patterns generated by the hepatocytes surrounding the vascular structures are very similar to those of ramified deterministic fractal objects obtained by the Diffusion-limited aggregation (DLA) process, which are examples of fractal aggregates in which individual particles represents the smallest aggregating unit and colonies represent an aggregate of particles. DLA process were originally developed by Witten and Sander to describe the morphogenesis of tree-like structures in non living systems using simple physical rules [3, 4].

Based upon considerations of these theoretical and experimental results we built a two-dimensional computer-liver in which the smallest aggregating unit (the single particle) is the hepatocyte, and colonies of hepatocytes represent an aggregate of particles.

Our aim was to purpose a deterministic growth model of the liver parenchyma in which the final morphological pattern resembles its visible in vivo structure, in normal conditions.

According to Dioguardi the formation of the liver parenchyma happens in two ways:

a. for centripetal attraction;

b. for duplication process (mitosis).

In our computer simulation, the first step is to define a seed, representing the central hepatic canals (CLV), usually at the centre of an empty system. Diffusing hepatocytes of the same size are then released into the system from a random point far from the seed, and are allowed to move until they arrive at the vicinity of the existing hepatocytes, at which time they stop their diffusion and adhere to the system. There is not limit to the quantity of CLV aggregation points. It is also possible choose the dimensions of the hepatocytes.

We have considered also the possibility that hepatocytes could duplicate. Cell duplication may only evolve in a radial pattern but this duplication is possible only if there is sufficient space between two cells.

In conclusion, our computer simulation is a useful tool for explaining a complex system (such as liver parenchyma), and an essential step in evolution of theory from observation. Finally, computer models are likely to play an increasing role in the progress of biomedical research.


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1. Dioguardi N. Fegato a pi dimensioni. Etas Libri, 1992.

2. Dioguardi N. The liver, the hepatone and its functioning. Videotape published by the Centro di Medicina Teoretica, University of Milan, Italy, 1996.

3. Witten TA, Sander LM. Diffusion-limited aggregation, a kinetic critical phenomenon. Phys Rev Lett 1981;47: 1400-1403.

4. Meakin P. A New model for biological pattern formation. J Theor Biol 1986;118: 101-113.


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