ABSTRACT
CONTENTS 7.1 Theory of Biological Systems: Introduction . . . . . . . . . . . . . . . . . . 336
7.1.1 Some Basic Systemic Concepts in Biology and Physiology . . 336 7.1.1.1 Systems, Elements, and Relations . . . . . . . . . . . . . . 336 7.1.1.2 Feedback and Self-Regulation . . . . . . . . . . . . . . . . . 337 7.1.1.3 Environment, Closed Systems, Open Systems, and
Homeostasis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 7.1.1.4 Entropy and Negentropy . . . . . . . . . . . . . . . . . . . . 340 7.1.1.5 Autopoiesis, Adaptation, and Cybernetics . . . . . . . . 342 7.1.1.6 A Hierarchy of Systems . . . . . . . . . . . . . . . . . . . . . 343
7.1.2 Mathematical Modeling and Analysis of Biological Systems: Population Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 7.1.2.1 The Growth of a Single Species’ Population . . . . . . 345 7.1.2.2 The Logistic Equation . . . . . . . . . . . . . . . . . . . . . . . 346 7.1.2.3 The Lotka-Volterra Predator-Prey Model . . . . . . . . 348 7.1.2.4 A General Model for the Interaction of Two
Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 7.1.2.5 Modeling Epidemic Phenomena . . . . . . . . . . . . . . . 351 7.1.2.6 Physiological Systems . . . . . . . . . . . . . . . . . . . . . . 352 7.1.2.7 Compartmental Analysis . . . . . . . . . . . . . . . . . . . . 357 7.1.2.8 State, State Space, and State Equations . . . . . . . . . . 358 7.1.2.9 How to Choose the State Variables . . . . . . . . . . . . . 364 7.1.2.10 Solution of the Linear State Equation in the Time
Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 7.1.2.11 Nonlinear Systems, Singular Points, and
Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 7.2 Tracer Kinetics Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
7.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 7.2.2 Concepts in Tracer Kinetics Modeling . . . . . . . . . . . . . . . . . 388 7.2.3 Transport of Tracers and Localization Mechanisms . . . . . . . 392 7.2.4 Compartmental Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
7.2.4.1 Single-Compartment Models and Multicompartment Models . . . . . . . . . . . . . . . . . . . 397
7.2.5 Tracers, Volumes, and Flows in Dilution Systems . . . . . . . . . 407
7.2.5.1 The Stewart-Hamilton Principle . . . . . . . . . . . . . . . 413 7.2.5.2 Volume Calculation . . . . . . . . . . . . . . . . . . . . . . . . 414
7.2.6 Distribution Systems and the Convolution Integral . . . . . . . 415 7.2.6.1 Determination of Flow . . . . . . . . . . . . . . . . . . . . . . 417
7.2.7 Regeneration of the Frequency Function of Transit Times by Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
7.2.8 Data Analysis and Models in PET Studies . . . . . . . . . . . . . . 422 7.2.9 Parametric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 7.2.10 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
7.1.1 Some Basic Systemic Concepts in Biology and Physiology
In this chapter, we will study the basic notions of systems theory that are most relevant to biological systems (including physiological ones). In the last few decades, the methods for modeling and analyzing biological systems have probably been one of the most relevant facets of the enormous development of mathematical biology.
