What Is a Model and Why Are They Made

A system is a collection of objects that interact to create a unified whole, such as a cell culture system, a rat, or a human. The type of models that are of interest in this book are mathematical models that represent the system of interest and can be used to explore the structure and behavior of the system (Wastney et al., 1997). A more simplistic definition might be that a mathematical model defines how you think your data were generated. Most famous mathematical models can be found in chemistry and physics, such as  

Mathematical models in biology tend to be more complex, but are all based on the same foundations used to develop models in the more physically oriented sciences. 

In defining a mathematical model it is helpful to distinguish between the various components of the model. Models are built using experimentally derived data. This so-called data generating process is dependent on system inputs, system dynamics, and the device used to measure the output from a system (Fig. 1.1). But in addition to these systematic processes are the sources of error that confound our measurements. These errors may be measurement errors but also include process noise that is part of the system. One goal of mathematical modeling is to differentiate the information or systematic component in the system from the noise or random components in the system, i.e., 

models usually consist of a structural model or systematic component plus a statistical model that describes the error component of the model. Early in the modeling process the focus may lie with the systematic component and then move to a more holistic approach involving the error components. For example, the 1-compartment model after bolus administration is 

The first term on the right hand side of Eq. (1.1) is the structural model having two inputs (also called independent variables), D (dose) and t (time), and one output (also called the dependent variable), C (concentration). The variables V (volume of distribution) and CL (clearance) are referred to as model parameters which must be estimated from the observed concentration data. The second term in Eq. (1.1) is the error component (also called the variance model), e represents the deviation between model predicted concentrations and observed concentrations. 

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