Solutions to the diffusion equation with solute binding and elimination

5 SOLUTIONS TO THE DIFFUSION EQUATION WITH SOLUTE BINDING AND ELIMINATION 

1 Diffusion with Reversible Binding to Immobilized Elements 

Consider a solute that diffuses within the extracellular space of the tissue, but also interacts with the tissue by reversibly binding to some fixed component of the tissue. For example, the diffusing solute might bind to a protein on the cell surface or to a protein in the extracellular matrix. When bound the solute may 

Upon substitution of the equilibrium relationship and the assumption that is constant with time, this equation becomes 

Therefore, the net effect of a homogeneous, rapid, reversible reaction is to retard the rate of diffusion of solute through the tissue. Solutions to this equation are identical to solutions of the pure diffusion equation (compare Equation 3-31 with Equation 3-52), except that the diffusion coefficient is reduced by a factor equal to the binding constant plus unity. These same equations can be used to evaluate penetration into tissues when more complicated equilibrium expressions are appropriate, by substituting the non-linear equilibrium expression into Equation 3-50 and solving the resulting equation (see [7]). 

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