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A mathematical model for targeting chemicals to tissues by exploiting complex degradation

Bruce S Gardiner1*, Lihai Zhang2, David W Smith1*, Peter Pivonka1 and Alan J Grodzinsky3

Author Affiliations

1 School of Computer Science and Software Engineering, The University of Western Australia, WA, 6009, Australia

2 Department of Infrastructure Engineering, University of Melbourne, VIC 3010, Australia

3 Center for Biomedical Engineering, Department of Electrical Engineering and Computer Science, and Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

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Biology Direct 2011, 6:46  doi:10.1186/1745-6150-6-46

Published: 22 September 2011



In many biological and therapeutic contexts, it is highly desirable to target a chemical specifically to a particular tissue where it exerts its biological effect. In this paper, we present a simple, generic, mathematical model that elucidates a general method for targeting a chemical to particular tissues. The model consists of coupled reaction-diffusion equations to describe the evolution within the tissue of the concentrations of three chemical species: a (concentration of free chemical), b (binding protein) and their complex, c (chemical bound to binding protein). We assume that all species are free to diffuse, and that a and b undergo a reversible reaction to form c. In addition, the complex, c, can be broken down by a process (e.g. an enzyme in the tissue) that results in the release of the chemical, a, which is then free to exert its biological action.


For simplicity, we consider a one-dimensional geometry. In the special case where the rate of complex formation is small (compared to the diffusion timescale of the species within the tissue) the system can be solved analytically. This analytic solution allows us to show how the concentration of free chemical, a, in the tissue can be increased over the concentration of free chemical at the tissue boundary. We show that, under certain conditions, the maximum concentration of a can occur at the centre of the tissue, and give an upper bound on this maximum level. Numerical simulations are then used to determine how the behaviour of the system changes when the assumption of negligible complex formation rate is relaxed.


We have shown, using our mathematical model, how complex degradation can potentially be exploited to target a chemical to a particular tissue, and how the level of the active chemical depends on factors such as the diffusion coefficients and degradation/production rates of each species. The biological significance of these results in terms of potential applications in cartilage tissue engineering and chemotherapy is discussed. In particular, we believe these results may be of use in determining the most promising prodrug candidates.

Solute transport; Tissue; Mathematical model; Prodrug; Insulin-like Growth factor (IGF); proteases