Application of a random network with a variable geometry of links to the kinetics of drug elimination in healthy and diseased livers

This paper discusses an application of a random network with a variable number of links and traps to the elimination of drug molecules from the body by the liver. The nodes and links represent the transport vessels, and the traps represent liver cells with metabolic enzymes that eliminate drug molecules. By varying the number and configuration of links and nodes, different disease states of the liver related to vascular damage have been simulated, and the effects on the rate of elimination of a drug have been investigated. Results of numerical simulations show the prevalence of exponential decay curves with rates that depend on the concentration of links. In the case of fractal lattices at the percolation threshold, we find that the decay of the concentration is described by exponential functions for high trap concentrations but transitions to stretched exponential behavior at low trap concentrations.

Figure 1

The liver. (a) The macrostructure. The hepatic artery and portal vein bring blood rich in oxygen, nutrients, and drug molecules in from the left. (b) The vessels branch to form a tree of arterioles and venules. (c) The terminal vessel (shaded circle) empties into an acinus. The small black circles are hepatocytes, and the white spaces consist of sinusoids and extracellular space. The three white circles on the periphery are collecting vessels that lead to the hepatic vein though a tree similar to the supply one pictured in (b). (d) The equivalent network representation. The molecules enter the lattice from the left through a supply vessel (shaded circle) and leave from the right by a collecting vessel (open circle). In between, they perform a random walk. The black circles represent hepatocytes and the empty nodes represent acini and extracellular space.

Figure 2

The network model interpretation for (a) a healthy liver, (b) a liver with hepatitis, (c) a liver with necrotic cirrhosis, and (d) a liver with vascular damage. The shaded circles represent supply nodes, the open circles represent collecting nodes, the empty nodes represent S sites, and the black nodes represent H sites.

Figure 3

A log-lin plot of the concentration as a function of time for various values of the probability $p$ that bonds are removed in the case of a $120\times 120$ lattice with 50 traps.

Figure 4

The relationship between the kinetic rate constant $k$ and the probability $p$.

Figure 5

Fit of Eq. (9) (solid lines) to the concentration data from computer simulations in the case of a percolation cluster with a low number and a high number of traps.