Stiff integration package

We will now illustrate the way that equation (2.70) could be solved as part of a stiff integration package. The solution relies partly on using the New-ton-Raphson technique for solving nonlinear simultaneous equations, the principles of which will now be explained. We may describe a system of n nonlinear. 

The system of differential equations is integrated using CVODE numerical integration package. CVODE is a solver for stiff and nonstiff ordinary differential equation systems [60]. The fraction of dose absorbed is calculated as the sum of all drug amounts crossing the apical membrane as a function of time, divided by the dose, or by the sum of all doses if multiple dosing is used. 

For a system of S chemical species and R reactions c is the S vector of concentrations, k the R vector of time independent parameters (rate coefficients), and f the vector of the R rate expression functions. If the overall reaction is isothermal and takes place in a well-mixed vessel, equation (1) comprises a detailed chemical kinetic model (DCKM) of the reaction. The integration of the model equations can present difficulties because the rate coefficients may vary from one another by many orders of magnitude, and the differential equations are stiff. Numerical methods for the solution of stiff equations are discussed by Kee et al. [1]. Efficient solvers for stiff sets of equations have been developed and are available in various software packages. Some of these are described in Chapter 5. Additional information can be found in Refs. [2,3]. 

Numerical integration techniques are necessary in modeling and simulation of batch and bio processing. In this chapter we described error and stability criteria for numerical techniques. Various numerical techniques for solution of stiff and non-stiff problems are discussed. These methods include one-step and multi-step explicit methods for non-stiff and implicit methods for stiff systems, and orthogonal collocation method for ordinary as well as partial differential equations. These methods are an integral part of some of the packages like MATLAB. However, it is important to know the theory so that appropriate method for simulation can be chosen. 

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