# Multi-Step Methods

Unlike the point-slope methods, the multi-step methods require more than one previous point but they allow even longer step sizes. Although the calculations are more involved they can be quite efficient for certain calculations. Figure 10.4.1 Linear plot of Cp versus time Illustrating a Multi-Step Method

The Adams-Bashford method uses the 0, -1, -2 and -3 data point to calculate the +1 value. Thus, quite large stepsizes can be used accurately. An additional advantage is that only one additional calculation is needed for each new step. The earlier points must be calculated with another method, such as the Runge-Kutta method. A refinement of this method is the Adams-Moulton method which includes a back calculation step to confirm the accuracy of the first step. These methods are called predictor-correcter methods (forward-backward calculations).

These method work well with ordinary differential equations. They are very useful for the most of problems encountered when integrating differential equations derived from compartmental pharmacokinetic models. However, there are occasions in pharmacokinetics where these methods become very inefficient. This situation is stiff systems. Stiff systems are models where the ratio between the fastest and slowest rates constants is greater than 500 (stiffness ratio > 500). That is kfast/kslow > 500. This is common with physiologically based pharmacokinetic models as described in Chapter 23.

Gear's method is an answer to this problem (1,2). Gear's method, a predictor-correcter method, is very efficient for stiff systems. It is quite capable of efficiently working with systems with a stiffness ratio greater than 106.

References
1. Gear, C.W. 1971 "Algorithm 407 DIFSUB for Solution of Ordinary Differential Equations", Comm. ACM., 14, pp 185-190
2. Gear, C.W. 1971 "The Automatic Integration of Ordinary Differential Equations", Comm. ACM., 14, pp 176-179