This paper addresses the fundamental problem of how to connect a heat generating volume to a point heat sink by using a finite amount of high-conductivity material that can be distributed through the volume. The problem is one of optimizing the access (or minimizing the thermal resistance) between a finite-size volume and one point. The solution is constructed by covering the volume with a sequence of building blocks, which proceeds toward larger sizes (assemblies), hence, the “constructal” name for this approach. Optimized numerically at each stage are geometric features such as the overall shape of the building block, its number of constituents, and the internal distribution of high-conductivity inserts. It is shown that in the optimal design, the high-conductivity material has a distribution with the shape of a tree. Every aspect of the tree architecture is deterministic: the shapes of the largest assembly and all its constituents, the number of branches at each level of assembly, the relative position of building blocks in each assembly, and the relative thicknesses of successive branches. The finer, innermost details of the tree architecture (e.g., the branching angle) have a negligible effect on the overall thermal resistance. The main conclusion is that the structure, working mechanism, and minimal resistance of the tree network can be obtained deterministically, and that the constrained optimization of access routes accounts for the macroscopic structure in nature.

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The incorporation of monolayers of cultured hepatocytes into an extracorporeal perfusion system has become a promising approach for the development of a temporary bioartificial liver (BAL) support system. In this paper we present a numerical investigation of the oxygen tension, shear stress, and pressure drop in a bioreactor for a BAL composed of plasma-perfused chambers containing monolayers of porcine hepatocytes. The chambers consist of microfabricated parallel disks with center-to-edge radial flow. The oxygen uptake rate (OUR), measured in vitro for porcine hepatocytes, was curve-fitted using Michaelis-Menten kinetics for simulation of the oxygen concentration profile. The effect of different parameters that may influence the oxygen transport inside the chambers, such as the plasma flow rate, the chamber height, the initial oxygen tension in the perfused plasma, the OUR, and K(m) was investigated. We found that both the plasma flow rate and the initial oxygen tension may have an important effect upon oxygen transport. Increasing the flow rate and/or the inlet oxygen tension resulted in improved oxygen transport to cells in the radial-flow microchannels, and allowed significantly greater diameter reactor without oxygen limitation to the hepatocytes. In the range investigated in this paper (10 microns < H < 100 microns), and for a constant plasma flow rate, the chamber height, H, had a negligible effect on the oxygen transport to hepatocytes. On the contrary, it strongly affected the mechanical stress on the cells that is also crucial for the successful design of the BAL reactors. A twofold decrease in chamber height from 50 to 25 microns produced approximately a fivefold increase in maximal shear stress at the inlet of the reactor from 2 to 10 dyn/cm2. Further decrease in chamber height resulted in shear stress values that are physiologically unrealistic. Therefore, the channel height needs to be carefully chosen in a BAL design to avoid deleterious hydrodynamic effects on hepatocytes.