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1. An Introduction to Modelling Methodology
1.1. Introduction
The aim of this book is to describe more recent developments in modelling theory and practice in relation to physiology and medicine. The chapters that follow offer detailed accounts of several facets of modelling methodology (Chapters 2-6) as well as demonstration of how such methodological development can be applied in areas of physiology and medicine. This application material, contained in Chapters 7-13, is not intended to be comprehensive. Rather, topics have been chosen that span study in the circulatory and respiratory systems (Chapters 8-10) and key areas of metabolism and endocrinology (Chapters 7 and I1-13). The authors of the respective chapters have very considerable expertise in these areas of physiology and medicine.
Before moving into the more advanced areas of methodology, it is appropriate to review the fundamentals of the modelling process, which put simply can be viewed as a mapping or transforming of a physiological system into a model as shown in Figure 1.1. The process has now reached substantial maturity, and the basic ingredients are well established. This overall modelling framework is described in detail in the accompanying volume (Cobelli and Carson, 2001). In this chapter, we provide a distillation of that framework and revisit the fundamentals upon which the later, more detailed chapters are built.
1.2. The Need for Models
1.2.1. Physiological Complexity
Complexity is what characterises much of physiology, and we must have a method to address this. Complexity manifests itself through elements that comprise any physiological system through the nature of their connectivity, in terms of hierarchy, and through the existence of nonlinear, stochastic, and time-varying effects. Complexity is exhibited at each level of the hierarchy and across levels within the physiological system.
What do we mean by some of these concepts? First, the physiological hierarchy will include the levels of molecule, cell, organ, and organism. Complex processes of regulation and control are evident at each level. Feedback is another key feature that occurs in many forms. It is inherent in chemical reactions within the organism. There are explicit hormonal control mechanisms superimposed upon metabolic processes. The physiological organ systems exhibit explicit control mechanisms. In many instances, there is negative feedback, although examples of positive feedback also exist. Feedback offers examples of control action being taken not only in relation to changes in the value of a physiological variable per se, but also in response either to its rate of change or to the integral of its value over a period of time. Some of these concepts of feedback and control are examined in more detail in Chapter 2.
As a result of this physiological complexity, it is not often possible to measure directly (in vivo) the quantities of interest. Only indirect measures may be feasible, implying the need for some model to be able to infer the value of the quantity of real interest. Measurement constraints usually mean that it is only possible to obtain readings of blood values of a metabolite when the real interest lies in its value in body tissue. Equally, it is not generally possible to measure the secretions of the endocrine glands.
Overall, this complexity-coupled with the limitations that are imposed upon the measurement processes in physiology and medicine-means that models must be adopted to aid our understanding.
1.2.2. Models and Their Purposes
What do we mean by the term model? In essence, it is a representation of reality involving some degree of approximation. Models can take many forms. They can be conceptual, mental, verbal, physical, statistical, mathematical, logical, or graphical in form. For the most part, this volume focuses on mathematical modelling.
Given that a model provides an approximate representation of reality, what is the purpose of modelling activity? As is shown in Figure 1.2, the purpose is a key driver of good modelling methodology. In classic scientific terms, modelling can be used to describe, interpret, predict, or explain. A mathematical expression, for example, a single exponential decay, can provide a compact description of data that approximate to a first-order process. A mathematical model can be used to interpret data collected as part of a lung function test. A model of renal function, which includes representations of the dynamics of urea and creatinine, can be used to predict the time at which a patient with end-stage renal disease should next undergo haemodialysis. A model of glucose and insulin can be used to gain additional insight into, and explanation of, the complex endocrine dynamics in the diabetic patient.
Rather, more specific purposes for modelling can be identified in the physiological context. These include aiding understanding, testing hypotheses, measuring inferences, teaching, simulating, and examining experimental design. For example, competing models, constituting alternative hypotheses, can be examined to determine which are compatible with physiological or clinical observation. Equally, a model of the relevant metabolic processes, when taken together with measurements of a metabolite made in the bloodstream, can be used to infer the value of that metabolite in the liver. Models also are increasingly used as a medium in teaching and learning processes, where, by means of simulation, the student can be exposed to a richer range of physiological and pathophysiological situations than would be possible in the conventional physiological laboratory setting. Models also can play a powerful role in experimental design. For instance, if the number of blood samples that can be withdrawn from a patient is limited in a given period of time, models can be used to determine the times at which blood samples should be withdrawn to obtain the maximum information from the experiment, for example, in relation to pharmacokinetic or pharmacodynamic effects.
Considering what is meant by a model and its purposes, we now focus on the nature of the process itself. As already indicated, this is the process of mapping from the physiological or pathophysiological system of interest to the completed model, as shown in Figure 1.1. The essential ingredients are model formulation, including determination of the degree to which the model is an approximation of reality; model identification, including parameter estimation; and model validation. These are discussed in the following sections.
1.3. Approaches to Modelling
In developing a mathematical model, two fundamental approaches are possible. The first is based on experimental data and is essentially a datadriven approach. The other is based on a fundamental understanding of the physical and chemical processes that give rise to the resultant experimental data. This can be referred to as modelling the system.
1.3.1. Modelling the Data
Models that are based on experimental data are generally known as datadriven or black box models. Fundamentally, this means seeking quantitative descriptions of physiological systems based on input-output (1/O) descriptions derived from experimental data collected on the system. Simply put, these are mathematical descriptions of data, with only implicit correspondence to the underlying physiology.
Why should we use such data models? First, they are particularly appropriate where there is a lack of knowledge of the underlying physiology, whether a priori knowledge or knowledge acquired directly through measurement. Equally, they are appropriate when an overall 1/O representation of the system's dynamics is needed, without knowing specifically how the physiological mechanisms gave rise to such 1/O behaviour.
The methodological framework for modelling data is depicted in Figure 1.3. Several specific methods are available for formulating such data models, including time series methods, transfer function analysis, convolutiondeconvolution techniques that are restricted to linear systems (discussed in Chapter 3), and impulse response methods...
