Mathematical and Computational Methods in Physiology: Satellite Symposium of the 28th International Congress of Physiological Sciences, Budapest, Hungary, 1980

Mathematical and Computational Methods in Physiology discusses the importance of quantitative description of physiological phenomena and for quantitative comparison of experimental data. An article explains the homeostasis of the body with a focus on the controlling aspects. This section evaluates the concepts of modern physiology and biocybernetics. The canal-ocular reflex and the otolith-ocular reflex in man stimulates eye rotations compensatory for head angular and linear displacements. The book enumerates some modelling and simulation to observe the visual-vestibular interaction during angular and linear body acceleration. A section on the determination of cardiovascular control is given. The text reviews the mathematical models of the biological age of the rat. A numerical simulation of water transport in epithelial junctions is explained comprehensively. A chapter analyzing the computer simulation of drug-receptor interaction is presented. The book will provide useful information to zoologists, doctors, ophthalmologists, students and researchers in the field of medicine.

• Language: English
• Publisher: Elsevier Science
• Release date: Oct 22, 2013
• ISBN: 9781483190228

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Modelling and simulation of physiological systems

Outline

Chapter 1: HOMEOSTASIS OF THE BODY: CONTROL ASPECTS

Chapter 2: VISUAL–VESTIBULAR INTERACTION DURING ANGULAR AND LINEAR BODY ACCELERATION: MODELLING AND SIMULATION

Chapter 3: A VOLTERRA-LIKE DISSECTION OF DYNAMIC TEST RESPONSES FOR THE IDENTIFICATION OF CARDIOVASCULAR CONTROL

Chapter 4: MODELLING OT THE CO2 REBREATHING CARDIAC OUTPUT MEASUREMENT

Chapter 5: PROBLEMS OF APPLYING BIOMECHANICAL MODELS IN DETERMINING CARDIOVASCULAR SYSTEM’S RELIABILITY DURING A COSMIC MISSION

Chapter 6: A MATHEMATICAL MODEL OF MECHANICAL PROPERTIES OF THE RAT’S AORTA

Chapter 7: MATHEMATICAL MODELS OF THE BIOLOGICAL AGE OF THE RAT

Chapter 8: NUMERICAL SIMULATION OF WATER TRANSPORT IN EPITHELIAL JUNCTIONS

Chapter 9: ANALYTIC PROPERTIES AND IDENTIFIABILITY PROBLEMS OF COMPARTMENTAL MODELS WITH TIME-LAGS

Chapter 10: THE MODELLING OF TAGGED PARTICLES MIGRATION IN PHYSIOLOGICAL SYSTEMS

Chapter 11: CRITERIA FOR COMPUTER SIMULATION OF DRUG–RECEPTOR INTERACTION

Chapter 12: A NEW STOCHASTIC APPROACH TO COMPARTMENTS WITH COMBINED CROSS-SECTION AND TIME-SERIES DATA

Chapter 13: STATIONARY DISTRIBUTIONS IN STOCHASTIC KINETICS

Chapter 14: OSCILLATORY PHENOMENA AT THE SYNAPSE

Chapter 15: SPATIAL GROWTH OF TUMORS. A SIMULATION STUDY

Chapter 16: SOME MODEL APPLICATIONS OF THE FUZZY SETS THEORY IN DECISION-MAKING

Chapter 17: COMPLEX METHOD FOR THE DETERMINATION OF THE PHYSIOLOGICAL PARAMETERS OF BACTERIUM–PHAGE SYSTEMS

Chapter 18: CONCLUDING REMARKS ON MODELLING AND SIMULATION OF PHYSIOLOGICAL SYSTEMS

HOMEOSTASIS OF THE BODY: CONTROL ASPECTS

V.N. Novoseltsev,     Institute of Control Sciences, Profsojuznaya 65, 117342, Moscow, USSR

Publisher Summary

Some essential problems in the investigation of control aspects of homeostasis as they can be seen by control engineer follow from the specific features of homeostatic behavior of physiological systems. The main feature of a modern treatment of homeostasis in terms of control theory is the desire to spread this concept to the whole homeostatic curve, not only for narrow plateau part. A mathematical description of a physiological regulatory system can be treated in terms of four main groups of variables. The functions of these variables in the model and the structure of the model itself depend on preferences and views of its designer. The control theory methods proved to be the general tool of investigation of all the regulatory features in physiological systems. Both theoretical and practical problems can be solved at present within the same framework of modern control theory methods.

One of the most important concepts of modern physiology and biocybernetics is the concept of homeostasis. The growing interest to the problem of homeostatic behavior is determined now by new problems in science and technology which arose during last years due to drastic spread of the life area of mankind and technological activity.

Some essential problems in investigation of control aspects of homeostasis as they can be seen by control engineer follow from the specific features of homeostatic behavior of physiological systems. In fig. 1 so called homeostatic (or regulatory) curve is shown, which is typical for the most of essential internal variables of the body when external parameters are changing.

Fig. 1 A typical homeostatic (regulatory) curve

In the central part the curve is flat, so it has so called plateau region, where a relative constancy of the internal variable is seen. The concept of homeostasis was firstly formulated for this rather narrow part of the curve. Homeostasis was treated as ability of the body to maintain all its essential variables within some predetermined limits.

The main feature of a modern treatment of homeostasis in terms of control theory is the desire to spread this concept to the whole homeostatic curve, not only for narrow plateau part. The following problems arise now:

1. How is homeostatic behaviour organized in terms of automatic control theory?

2. How can it be described mathematically and modelled?

3. How can it be measured?

4. How can homeostasis of the body maintained artificially?

THE MAIN FEATURES OF HOMEOSTATIC BEHAVIOUR

When a theoretician describes control processes in physiological systems he stresses the general principles of control paying a little attention to specific features of concrete physiological regulations. One can determine the two main types of variables common for all the regulatory systems. These are levels and rates. Levels describe the state of a system, rates - its activity [1].

It is needed to stress that in physiological regulatory systems it is not enough to use just these two kinds of signals. The key point is the rates are of two quite different types. Some of rates are not physiologically regulated, the values of these set by Nature itself and they are absolutely needed for life. Basal metabolic rate is an exellent example of this group of rates.

The another group includes the rates of physiologically regulated processes, which are under the control of physiological mechanisms. Blood flow rates, oxygen delivery rates, rate of perspiration are examples of this type of rates.

The rates of the first type can be termed initial (or primary), those of the other type – response (or secondary) rates.

Now all the processes in a physiological regulatory system can be treated in terms of four main groups of variables -

w - initial rates (the rates of primary processes, which are not physiologically,

y - response rates (the rates of secondary processes, which can be controlled by physiological regulators/,

x - levels of substances or energy inside the system,

v - levels of substances outside the system.

For example, in fig. 2 the rates of processes and levels in an oxygen delivery and consumption system are shown and denominated.

Fig. 2 Distribution of signals in an oxygen transport and consumption system

STRUCTURE OF THE MODEL

Usually a mathematical description of a physiological regulatory system can be treated in terms of four main groups of variables given above. Nevertheless the functions of these variables in the model and the structure of the model itself depends on preferences and views of its designer.

The structure of the model is determined by the answer to the next question: what kind of signals, levels or rates, is to be controlled in a physiological system?

From one hand all the factors of physiological knowledge tell rates are to be controlled. Rates are under physiological control to meet demands of a body in substances and energy, to keep its thermodynamical stationary non-equilibrium state. As early as 1860 this idea was declared by the great Russian scientist Ivan Setchenov [2].

From the other hand there is an obvious constancy of all the essential variables which can be only due to some sort of control. This idea was expressed first by Claude Bernard [3] and then formulated as a concept of homeostatic behavior by W. Cannon [4]. This concept was embodied by N. Wiener [5] as a single-loop feedback control system.

In fig. 3 the two possible forms of organizing of physiological models are shown in terms of x, y, w and v variables.

Fig. 3 Classical model with control of levels (right) and modern model with control of rates (left)

The left one corresponds to the classical Wiener stucture where levels are considered to be controlled signals (L-model) The right one is the controlled rate model (R-model), which is an implementation of Setchenov’s ideas [6].

These models drastically differ in the distribution of the variables in the structure of control system as it is shown in table 1.

Table 1

The role of main groups of variables

Classical approach is formulated usually in terms of input-output, whereas R-model corresponds to modern input-output-state description [7].

It is clear from the fig. 4 that controlled rate R-model has a wider area of adequacy to its physiological prototype. The L-model approach is to be used basically for only the plateau range of homeostatic carve (see fig. 1), whereas R-models can be used under all possible variations of external conditions.

Fig. 4 Comparison of areas where classical models and modern control theory models can be used: v1, v2 - ambient parameters values

Some new fields of possible applications in applied physiology as well as in medical and technical problems connected with it are shown in fig. 5.

Fig. 5 New fields of practical applications of the models of physiological control systems. In both cases non-homeostatic behavior of homeostatic systems is to be investigated

is doubtful.

MATHEMATICAL DESCRIPTION OF R-MODELS

Now let us have a look at mathematical description of controlled rate model. Any practical problem in physiology and biocybernetics usually leads to rather complex nonlinear models, but here we discuss only a simple linear model. The block-diagram of such a model is as usual for state variable approach in modern control theory (see fig. 6). The only thing to be underlined is the strict correspondence of main groups of variables mentioned above to control theory signals: w is a reference input signal, x - state of the system, v – a disturbance input signal and y is an output signal. All the variables are vectors, so the model is described by usual input-outut-state equations:

Fig. 6 Block-diagram of controlled rate model in terms of modern control theory

(1)

where w is n-vector, y is r-vector, x is m-vector, v is 1-vector and A,B,C,D and P are matrices of corresponding dimensions [7].

Let us see how the main physiological features of a system can be described in terms of R-models.

Stationary non-equilibrium state is achieved when all the processed are balansed in a system. Any of initial rates in the model is compensated by one or more than one response rates:

(2)

where kij -coefficients (some of them may be zeros). To be exact,

(3)

R is ‘the matrix of controlled rates’, which can be expressed in terms of the matrices of the system:

(4)

Under this condition all the state variables of the system became constant:

(5)

Homeostasis is now to be treated as low sensitivity of state variables to external variations during stationary non-equilibrium. Denoting

(6)

one can characterize a homeostatic behavior of i-th variable xi under vj variations