The mathematics behind modeling

Abstract

The purpose of this chapter has been to furnish insight into the theoretical background on which compartmental modeling software packages are based. To accomplish this goal, only the basic ideas were stressed, avoiding discussion of the intricacies required for efficiency. The object was to remove the mystery from these powerful programs by examining the fundamental ideas which make them tick. The first section was concerned with how compartmental models are built up and how to obtain information concerning the system behavior described by these models. The cornerstone here is to describe a system by determining how it behaves over (typically) very short time periods. This leads to a differential equation description of a compartmental system. Information can be extracted from these equations by returning to their basic meaning, illustrated in their derivation. Computers are ideal for obtaining this information by piecing together the results obtained over short time periods, to find the behavior over long time periods. In an actual situation governed by a compartmental model, it's often the case that we may know only the form of the model, but not the values of the rate constants which must be known for its effective use. The second section of the chapter was devoted to the practical problem of determining the rate constants of the model, based on observed data. This is a matter of searching for those values which, in some sense, best fit the data. To attack this problem we need a reasonable criterion to judge how well a proposed model fits the data. We chose to use the total squared deviation, ψ, which is the most common such criterion - but not the only reasonable one. The search technique we examined - steepest descent - is based on a simple idea: looking at the total squared deviation criterion geometrically. In graphical terms, the best fit corresponds to finding the low point on a surface, whose height above any point at sea level is computable. If we could imagine the view of someone trying to find the low point from some arbitrarily chosen initial position on this mountain-like surface, we would look around and find the direction where (close by) the mountain drops off most steeply. We would go in that specific direction until we reach a low point, moving only along this initially chosen direction. At this new low point, we could change again to a direction of steepest descent, and keep repeating this process until we make no further effective downward progress. No guarantee in general is made for this process, but it often works. Finally, having found the best values for the rate constants, we must recognize that if the data is affected by factors not explicitly taken into account in the model, the variability induced by these factors precludes a perfect fit. For this reason, it is finally necessary to determine how good the model is, as a description of the data, and how accurate are the fitted rate constants. For the model fit, the sample RMS error (simply related to the total squared deviation, and often the same as the sample standard deviation) may be used. For determining the accuracy of the fitted rate constants, practical methods based on computer software simulations are recommended.